document_title: Pythagorean triple

question_text: pythagorean triplet in which one number is 12 is

question_tokens: pythagorean triplet in which one number is 12 is

root

text: Pythagorean triple

text: Examples[edit]

text: There are 16 primitive Pythagorean triples with c ≤ 100:

tag: p

text:
(3, 4, 5)
(5, 12, 13)
(8, 15, 17)
(7, 24, 25)

(20, 21, 29)
(12, 35, 37)
(9, 40, 41)
(28, 45, 53)

(11, 60, 61)
(16, 63, 65)
(33, 56, 65)
(48, 55, 73)

(13, 84, 85)
(36, 77, 85)
(39, 80, 89)
(65, 72, 97)

tag: table

text: Note, for example, that (6, 8, 10) is not a primitive Pythagorean triple, as it is a multiple of (3, 4, 5). Each of these low-c points forms one of the more easily recognizable radiating lines in the scatter plot.

text: Additionally these are all the primitive Pythagorean triples with 100 < c ≤ 300:

text:
(20, 99, 101)
(60, 91, 109)
(15, 112, 113)
(44, 117, 125)

(88, 105, 137)
(17, 144, 145)
(24, 143, 145)
(51, 140, 149)

(85, 132, 157)
(119, 120, 169)
(52, 165, 173)
(19, 180, 181)

(57, 176, 185)
(104, 153, 185)
(95, 168, 193)
(28, 195, 197)

(84, 187, 205)
(133, 156, 205)
(21, 220, 221)
(140, 171, 221)

(60, 221, 229)
(105, 208, 233)
(120, 209, 241)
(32, 255, 257)

(23, 264, 265)
(96, 247, 265)
(69, 260, 269)
(115, 252, 277)

(160, 231, 281)
(161, 240, 289)
(68, 285, 293)

text: Generating a triple[edit]

text: Euclid's formula[3] is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. The formula states that the integers

text: form a Pythagorean triple. The triple generated by Euclid's formula is primitive if and only if m and n are coprime and not both odd. When both m and n are odd, then a, b, and c will be even, and the triple will not be primitive; however, dividing a, b, and c by 2 will yield a primitive triple when m and n are coprime and both odd.[4]

text: Every primitive triple arises (after the exchange of a and b, if a is even) from a unique pair of coprime numbers m, n, one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of a, b and c to m and n from Euclid's formula is referenced throughout the rest of this article.

text: Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, (9, 12, 15) cannot be generated using integer m and n. This can be remedied by inserting an additional parameter k to the formula. The following will generate all Pythagorean triples uniquely:

text: where m, n, and k are positive integers with m > n, and with m and n coprime and not both odd.

text: That these formulas generate Pythagorean triples can be verified by expanding a2 + b2 using elementary algebra and verifying that the result equals c2. Since every Pythagorean triple can be divided through by some integer k to obtain a primitive triple, every triple can be generated uniquely by using the formula with m and n to generate its primitive counterpart and then multiplying through by k as in the last equation.

text: Many formulas for generating triples with particular properties have been developed since the time of Euclid.

text: Proof of Euclid's formula[edit]

text: That satisfaction of Euclid's formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers m and n, m > n, the a, b, and c given by the formula are all positive integers, and from the fact that

text: A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows.[5] All such triples can be written as (a, b, c) where a2 + b2 = c2 and a, b, c are coprime. Thus a, b, c are pairwise coprime (if a prime number would divide two of them, it would divide also the third one). As a and b are coprime, one is odd, and one may suppose that it is a, by exchanging, if needed, a and b. This implies that b is even and c is odd (if b were odd, c would be even, and c2 would be a multiple of 4, while a2 + b2 would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4).

text:
From

a

2

+

b

2

=

c

2

{\displaystyle a^{2}+b^{2}=c^{2}}

 we obtain

c

2

−

a

2

=

b

2

{\displaystyle c^{2}-a^{2}=b^{2}}

 and hence

(
c
−
a
)
(
c
+
a
)
=

b

2

{\displaystyle (c-a)(c+a)=b^{2}}

. Then

(
c
+
a
)

b

=

b

(
c
−
a
)

{\displaystyle {\tfrac {(c+a)}{b}}={\tfrac {b}{(c-a)}}}

. Since

(
c
+
a
)

b

{\displaystyle {\tfrac {(c+a)}{b}}}

 is rational, we set it equal to

m
n

{\displaystyle {\tfrac {m}{n}}}

 in lowest terms. Thus

(
c
−
a
)

b

=

n
m

{\displaystyle {\tfrac {(c-a)}{b}}={\tfrac {n}{m}}}

, as being the reciprocal of

(
c
+
a
)

b

{\displaystyle {\tfrac {(c+a)}{b}}}

. Then solving

text:
for

c
b

{\displaystyle {\tfrac {c}{b}}}

 and

a
b

{\displaystyle {\tfrac {a}{b}}}

 gives

text:
As

m
n

{\displaystyle {\tfrac {m}{n}}}

 is fully reduced, m and n are coprime, and they cannot be both even. If they were both odd, the numerator of

m

2

−

n

2

2
m
n

{\displaystyle {\tfrac {m^{2}-n^{2}}{2mn}}}

 would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2mn would not be a multiple of 4, implying that a would be even. Thus one of m and n is odd and the other is even, and the numerators of the two fractions with denominator 2mn are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of m and n and not the other; thus it does not divide m2 ± n2). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula

text: A longer but more commonplace proof is given in Maor (2007)[6] and Sierpiński (2003).[7]:4–7

text: Interpretation of parameters in Euclid's formula[edit]

text:
Suppose the sides of a Pythagorean triangle have lengths m2 − n2, 2mn, and m2 + n2, and suppose the angle between the leg of length m2 − n2 and the hypotenuse of length m2 + n2 is denoted as β. Then

tan

β
2

=

n
m

{\displaystyle \tan {\tfrac {\beta }{2}}={\tfrac {n}{m}}}

 and the full-angle trigonometric values are

sin

β

=

2
m
n

m

2

+

n

2

{\displaystyle \sin {\beta }={\tfrac {2mn}{m^{2}+n^{2}}}}

,

cos

β

=

m

2

−

n

2

m

2

+

n

2

{\displaystyle \cos {\beta }={\tfrac {m^{2}-n^{2}}{m^{2}+n^{2}}}}

, and

tan

β

=

2
m
n

m

2

−

n

2

{\displaystyle \tan {\beta }={\tfrac {2mn}{m^{2}-n^{2}}}}

.[8]

text: A variant[edit]

text: The following variant of Euclid's formula is sometimes more convenient, as being more symmetric in m and n (same parity condition on m and n).

text: If m and n are two odd integers such that m > n, then

text: are three integers that form a Pythagorean triple, which is primitive if and only if m and n are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of a and b, if a is even) from a unique pair m > n > 0 of coprime odd integers.

text: Elementary properties of primitive Pythagorean triples[edit]

text: General properties[edit]

text: The properties of a primitive Pythagorean triple (a, b, c) with a < b < c (without specifying which of a or b is even and which is odd) include:

text:
(
c
−
a
)
(
c
−
b
)

2

{\displaystyle {\tfrac {(c-a)(c-b)}{2}}}

 is always a perfect square.[9] As it is only a necessary condition but not a sufficient one, it can be used in checking if a given triple of numbers is not a Pythagorean triple when they fail the test. For example, the triple {6, 12, 18} passes the test that (c − a)(c − b)/2 is a perfect square, but it is not a Pythagorean triple.
When a triple of numbers a, b and c forms a primitive Pythagorean triple, then (c minus the even leg) and one-half of (c minus the odd leg) are both perfect squares; however this is not a sufficient condition, as the numbers {1, 8, 9} pass the perfect squares test but are not a Pythagorean triple since 12 + 82 ≠ 92.
At most one of a, b, c is a square.[10]
The area of a Pythagorean triangle cannot be the square[11]:p. 17 or twice the square[11]:p. 21 of a natural number.
Exactly one of a, b is odd; c is odd.[12]
Exactly one of a, b is divisible by 3.[7]:23–25
Exactly one of a, b is divisible by 4.[7]
Exactly one of a, b, c is divisible by 5.[7]
The largest number that always divides abc is 60.[13]
All prime factors of c are primes of the form 4n + 1.[14] Therefore c is of the form 4n + 1.
The area (K = ab/2) is a congruent number[15] divisible by 6.
In every Pythagorean triple, the radius of the incircle and the radii of the three excircles are natural numbers. Specifically, for a primitive triple the radius of the incircle is r = n(m − n), and the radii of the excircles opposite the sides m2 − n2, 2mn, and the hypotenuse m2 + n2 are respectively m(m − n), n(m + n), and m(m + n).[16]
As for any right triangle, the converse of Thales' theorem says that the diameter of the circumcircle equals the hypotenuse; hence for primitive triples the circumdiameter is m2 + n2, and the circumradius is half of this and thus is rational but non-integer (since m and n have opposite parity).
When the area of a Pythagorean triangle is multiplied by the curvatures of its incircle and 3 excircles, the result is four positive integers w > x > y > z, respectively. Integers −w, x, y, z satisfy Descartes's Circle Equation.[17] Equivalently, the radius of the outer Soddy circle of any right triangle is equal to its semiperimeter. The outer Soddy center is located at D, where ACBD is a rectangle, ACB the right triangle and AB its hypotenuse.[17]:p. 6
There are no Pythagorean triples in which the hypotenuse and one leg are the legs of another Pythagorean triple; this is one of the equivalent forms of Fermat's right triangle theorem.[11]:p. 14
Each primitive Pythagorean triangle has a ratio of area to squared semiperimeter that is unique to itself and is given by[18]

tag: ul

text:
No primitive Pythagorean triangle has an integer altitude from the hypotenuse; that is, every primitive Pythagorean triangle is indecomposable.[19]
The set of all primitive Pythagorean triples forms a rooted ternary tree in a natural way; see Tree of primitive Pythagorean triples.
Neither of the acute angles of a Pythagorean triangle can be a rational number of degrees.[20] (This follows from Niven's theorem.)

text: Special cases[edit]

text: In addition, special Pythagorean triples with certain additional properties can be guaranteed to exist:

text:
Every integer greater than 2 that is not congruent to 2 mod 4 (in other words, every integer greater than 2 which is not of the form 4n + 2) is part of a primitive Pythagorean triple.
Every integer greater than 2 is part of a primitive or non-primitive Pythagorean triple. For example, the integers 6, 10, 14, and 18 are not part of primitive triples, but are part of the non-primitive triples (6, 8, 10), (14, 48, 50) and (18, 80, 82).
There exist infinitely many Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly one (such triples are necessarily primitive). One method to generate such triples is the relation (2n + 1)2 + [2n(n + 1)]2 = [2n(n + 1) + 1]2, leading to triples (3,4,5), (5,12,13), (7,24,25), etc. More generally, for every odd integer j, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the even leg differ by j2.
There exist infinitely many primitive Pythagorean triples in which the hypotenuse and the longer of the two legs differ by exactly two. Generalization: For every integer k > 0, there exist infinitely many primitive Pythagorean triples in which the hypotenuse and the odd leg differ by 2k2.
There exist infinitely many Pythagorean triples in which the two legs differ by exactly one. For example, 202 + 212 = 292; these are generated by Euclid's formula when

m
−
n

n

{\displaystyle {\tfrac {m-n}{n}}}

 is a convergent to √2.
For each natural number n, there exist n Pythagorean triples with different hypotenuses and the same area.
For each natural number n, there exist at least n different Pythagorean triples with the same leg a, where a is some natural number
For each natural number n, there exist at least n different Pythagorean triples with the same hypotenuse.[7]:31
There exist infinitely many Pythagorean triples with square numbers for both the hypotenuse c and the sum of the legs a + b. According to Fermat, the smallest such triple[21] has sides a = 4,565,486,027,761; b = 1,061,652,293,520; and c = 4,687,298,610,289. Here a + b = 2,372,1592 and c = 2,165,0172. This is generated by Euclid's formula with parameter values m = 2,150,905 and n = 246,792.
There exist non-primitive Pythagorean triangles with integer altitude from the hypotenuse.[22][23] Such Pythagorean triangles are known as decomposable since they can be split along this altitude into two separate and smaller Pythagorean triangles.[19]

text: Geometry of Euclid's formula[edit]

text: Rational points on a unit circle[edit]

text: Euclid's formula for a Pythagorean triple

text: can be understood in terms of the geometry of rational points on the unit circle (Trautman 1998).

text: In fact, a point in the Cartesian plane with coordinates (x, y) belongs to the unit circle if x2 + y2 = 1. The point is rational if x and y are rational numbers, that is, if there are coprime integers a, b, c such that

text: By multiplying both members by c2, one can see that the rational points on the circle are in one-to-one correspondence with the primitive Pythagorean triples.

text: The unit circle may also be defined by a parametric equation

text: Euclid's formula for Pythagorean triples means that, except for (−1, 0), a point on the circle is rational if and only if the corresponding value of t is a rational number.

text: Stereographic approach[edit]

text: There is therefore a correspondence between points on the unit circle with rational coordinates and primitive Pythagorean triples. At this point, Euclid's formulae can be derived either by methods of trigonometry or equivalently by using the stereographic projection.

text: For the stereographic approach, suppose that P′ is a point on the x-axis with rational coordinates

text: Then, it can be shown by basic algebra that the point P has coordinates

text: This establishes that each rational point of the x-axis goes over to a rational point of the unit circle. The converse, that every rational point of the unit circle comes from such a point of the x-axis, follows by applying the inverse stereographic projection. Suppose that P(x, y) is a point of the unit circle with x and y rational numbers. Then the point P′ obtained by stereographic projection onto the x-axis has coordinates

text: which is rational.

text: In terms of algebraic geometry, the algebraic variety of rational points on the unit circle is birational to the affine line over the rational numbers. The unit circle is thus called a rational curve, and it is this fact which enables an explicit parameterization of the (rational number) points on it by means of rational functions.

text: Pythagorean triangles in a 2D lattice[edit]

text:
A 2D lattice is a regular array of isolated points where if any one point is chosen as the Cartesian origin (0, 0), then all the other points are at (x, y) where x and y range over all positive and negative integers. Any Pythagorean triangle with triple (a, b, c) can be drawn within a 2D lattice with vertices at coordinates (0, 0), (a, 0) and (0, b). The count of lattice points lying strictly within the bounds of the triangle is given by

(
a
−
1
)
(
b
−
1
)
−
gcd

(
a
,
b
)

+
1

2

;

{\displaystyle {\tfrac {(a-1)(b-1)-\gcd {(a,b)}+1}{2}};}

[24] for primitive Pythagorean triples this interior lattice count is

(
a
−
1
)
(
b
−
1
)

2

.

{\displaystyle {\tfrac {(a-1)(b-1)}{2}}.}

 The area (by Pick's theorem equal to one less than the interior lattice count plus half the boundary lattice count) equals

a
b

2

{\displaystyle {\tfrac {ab}{2}}}

 .

text: The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides (20, 21, 29), (12, 35, 37) and common area 210 (sequence A093536 in the OEIS). The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with (18108, 252685, 253333), (28077, 162964, 165365) and interior lattice count 2287674594 (sequence A225760 in the OEIS). Three primitive Pythagorean triples have been found sharing the same area: (4485,  5852,  7373), (3059, 8580, 9109), (1380, 19019, 19069) with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count.

text: Enumeration of primitive Pythagorean triples[edit]

text:
By Euclid's formula all primitive Pythagorean triples can be generated from integers

m

{\displaystyle m}

 and

n

{\displaystyle n}

 with

m
>
n
>
0

{\displaystyle m>n>0}

,

m
+
n

{\displaystyle m+n}

 odd and

gcd
(
m
,
n
)
=
1

{\displaystyle \gcd(m,n)=1}

. Hence there is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where

n
m

{\displaystyle {\tfrac {n}{m}}}

 is in the interval

(
0
,
1
)

{\displaystyle (0,1)}

 and

m
+
n

{\displaystyle m+n}

 odd.

text:
The reverse mapping from a primitive triple

(
a
,
b
,
c
)

{\displaystyle (a,b,c)}

 where

c
>
b
>
a
>
0

{\displaystyle c>b>a>0}

 to a rational

n
m

{\displaystyle {\tfrac {n}{m}}}

 is achieved by studying the two sums

a
+
c

{\displaystyle a+c}

 and

b
+
c

{\displaystyle b+c}

. One of these sums will be a square that can be equated to

(
m
+
n

)

2

{\displaystyle (m+n)^{2}}

 and the other will be twice a square that can be equated to

2

m

2

{\displaystyle 2m^{2}}

. It is then possible to determine the rational

n
m

{\displaystyle {\tfrac {n}{m}}}

.

text:
In order to enumerate primitive Pythagorean triples the rational can be expressed as an ordered pair

(
n
,
m
)

{\displaystyle (n,m)}

 and mapped to an integer using a pairing function such as Cantor's pairing function. An example can be seen at (sequence A277557 in the OEIS). It begins

text: Spinors and the modular group[edit]

text: Pythagorean triples can likewise be encoded into a matrix of the form

text: A matrix of this form is symmetric. Furthermore, the determinant of X is

text: which is zero precisely when (a,b,c) is a Pythagorean triple. If X corresponds to a Pythagorean triple, then as a matrix it must have rank 1.

text: Since X is symmetric, it follows from a result in linear algebra that there is a column vector ξ = [m n]T such that the outer product

text:
X
=
2

[

m

n

]

[
m

n
]
=
2
ξ

ξ

T

{\displaystyle X=2{\begin{bmatrix}m\\n\end{bmatrix}}[m\ n]=2\xi \xi ^{T}\,}

(1)

text: holds, where the T denotes the matrix transpose. The vector ξ is called a spinor (for the Lorentz group SO(1, 2)). In abstract terms, the Euclid formula means that each primitive Pythagorean triple can be written as the outer product with itself of a spinor with integer entries, as in (1).

text: The modular group Γ is the set of 2×2 matrices with integer entries

text: with determinant equal to one: αδ − βγ = 1. This set forms a group, since the inverse of a matrix in Γ is again in Γ, as is the product of two matrices in Γ. The modular group acts on the collection of all integer spinors. Furthermore, the group is transitive on the collection of integer spinors with relatively prime entries. For if [m n]T has relatively prime entries, then

text: where u and v are selected (by the Euclidean algorithm) so that mu + nv = 1.

text: By acting on the spinor ξ in (1), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if A is a matrix in Γ, then

text:
2
(
A
ξ
)
(
A
ξ

)

T

=
A
X

A

T

{\displaystyle 2(A\xi )(A\xi )^{T}=AXA^{T}\,}

(2)

text: gives rise to an action on the matrix X in (1). This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. It is convenient at this point (per Trautman 1998) to call a triple (a,b,c) standard if c > 0 and either (a,b,c) are relatively prime or (a/2,b/2,c/2) are relatively prime with a/2 odd. If the spinor [m n]T has relatively prime entries, then the associated triple (a,b,c) determined by (1) is a standard triple. It follows that the action of the modular group is transitive on the set of standard triples.

text: Alternatively, restrict attention to those values of m and n for which m is odd and n is even. Let the subgroup Γ(2) of Γ be the kernel of the group homomorphism

text: where SL(2,Z2) is the special linear group over the finite field Z2 of integers modulo 2. Then Γ(2) is the group of unimodular transformations which preserve the parity of each entry. Thus if the first entry of ξ is odd and the second entry is even, then the same is true of Aξ for all A ∈ Γ(2). In fact, under the action (2), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples (Alperin 2005).

text: The group Γ(2) is the free group whose generators are the matrices

text: Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices U and L.

text: Parent/child relationships[edit]

text: By a result of Berggren (1934), all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three linear transformations T1, T2, T3 below, where a, b, c are sides of a triple:

text:
new side a
new side b
new side c

T1:
a − 2b + 2c
2a − b + 2c
2a − 2b + 3c

T2:
a + 2b + 2c
2a + b + 2c
2a + 2b + 3c

T3:
−a + 2b + 2c
−2a + b + 2c
−2a + 2b + 3c

text: In other words, every primitive triple will be a "parent" to three additional primitive triples. Starting from the initial node with a = 3, b = 4, and c = 5, the operation T1 produces the new triple

text: and similarly T2 and T3 produce the triples (21, 20, 29) and (15, 8, 17).

text: The linear transformations T1, T2, and T3 have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of x2 + y2 − z2 over the integers.[25]

text: Relation to Gaussian integers[edit]

text: Alternatively, Euclid's formulae can be analyzed and proven using the Gaussian integers.[26] Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers and i is the square root of negative one. The units of Gaussian integers are ±1 and ±i. The ordinary integers are called the rational integers and denoted as Z. The Gaussian integers are denoted as Z[i]. The right-hand side of the Pythagorean theorem may be factored in Gaussian integers:

text: A primitive Pythagorean triple is one in which a and b are coprime, i.e., they share no prime factors in the integers. For such a triple, either a or b is even, and the other is odd; from this, it follows that c is also odd.

text: The two factors z := a + bi and z* := a − bi of a primitive Pythagorean triple each equal the square of a Gaussian integer. This can be proved using the property that every Gaussian integer can be factored uniquely into Gaussian primes up to units.[27] (This unique factorization follows from the fact that, roughly speaking, a version of the Euclidean algorithm can be defined on them.) The proof has three steps. First, if a and b share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume a = gu and b = gv with Gaussian integers g, u and v and g not a unit. Then u and v lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer h. But then the integer gh ≠ ±1 divides both a and b.) Second, it follows that z and z* likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor δ would also divide z + z* = 2a and z − z* = 2ib. Since a and b are coprime, that implies that δ divides 2 = (1 + i)(1 − i) = i(1 − i)2. From the formula c2 = zz*, that in turn would imply that c is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since c2 is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since z and z* share no prime factors, this doubling is also true for them. Hence, z and z* are squares.

text: Thus, the first factor can be written

text: The real and imaginary parts of this equation give the two formulas:

text: For any primitive Pythagorean triple, there must be integers m and n such that these two equations are satisfied. Hence, every Pythagorean triple can be generated from some choice of these integers.

text: As perfect square Gaussian integers[edit]

text: If we consider the square of a Gaussian integer we get the following direct interpretation of Euclid's formulae as representing a perfect square Gaussian integers.

text:
Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p

|

p

|

2

{\displaystyle |p|^{2}}

 is always a square it is possible to show that a Pythagorean triples correspond to the square of a prime Gaussian integer if the hypotenuse is prime.

text:
If the Gaussian integer is not prime then it is the product of two Gaussian integers p and q with

|

p

|

2

{\displaystyle |p|^{2}}

 and

|

q

|

2

{\displaystyle |q|^{2}}

 integers. Since magnitudes multiply in the Gaussian integers, the product must be

|

p

|

|

q

|

{\displaystyle |p||q|}

, which when squared to find a Pythagorean triple must be composite. The contrapositive completes the proof.

text: Relation to ellipses with integral dimensions[edit]

text: With reference to the figure and the definition of the foci of an ellipse, F1 and F2, for any point P on the ellipse, F1P + PF2 is constant.

text: As points A and B are both on the ellipse, F1A + AF2 = F1B + BF2. Due to symmetry, F1A + AF2 = F2A' + AF2 = AA' = 2 AC, and F1B + BF2 = 2 BF2. Hence, AC = BF2.

text: Thus, if BCF2 is a right-angle triangle with integral sides, the separation of the foci, linear eccentricity, minor axis and major axis are all also integers.[28]

text: Distribution of triples[edit]

text: There are a number of results on the distribution of Pythagorean triples. In the scatter plot, a number of obvious patterns are already apparent. Whenever the legs (a,b) of a primitive triple appear in the plot, all integer multiples of (a,b) must also appear in the plot, and this property produces the appearance of lines radiating from the origin in the diagram.

text: Within the scatter, there are sets of parabolic patterns with a high density of points and all their foci at the origin, opening up in all four directions. Different parabolas intersect at the axes and appear to reflect off the axis with an incidence angle of 45 degrees, with a third parabola entering in a perpendicular fashion. Within this quadrant, each arc centered on the origin shows that section of the parabola that lies between its tip and its intersection with its semi-latus rectum.

text:
These patterns can be explained as follows. If

a

2

/

4
n

{\displaystyle a^{2}/4n}

 is an integer, then (a,

|

n
−

a

2

/

4
n

|

{\displaystyle |n-a^{2}/4n|}

,

n
+

a

2

/

4
n

{\displaystyle n+a^{2}/4n}

) is a Pythagorean triple. (In fact every Pythagorean triple (a, b, c) can be written in this way with integer n, possibly after exchanging a and b, since

n
=
(
b
+
c
)

/

2

{\displaystyle n=(b+c)/2}

 and a and b cannot both be odd.) The Pythagorean triples thus lie on curves given by

b
=

|

n
−

a

2

/

4
n

|

{\displaystyle b=|n-a^{2}/4n|}

, that is, parabolas reflected at the a-axis, and the corresponding curves with a and b interchanged. If a is varied for a given n (i.e. on a given parabola), integer values of b occur relatively frequently if n is a square or a small multiple of a square. If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, 382 = 1444, 2 × 272 = 1458, 3 × 222 = 1452, 5 × 172 = 1445 and 10 × 122 = 1440; the corresponding parabolic strip around n ≈ 1450 is clearly visible in the scatter plot.

text: The angular properties described above follow immediately from the functional form of the parabolas. The parabolas are reflected at the a-axis at a = 2n, and the derivative of b with respect to a at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value 2n also corresponds to a cluster. The corresponding parabola intersects the b-axis at right angles at b = 2n, and hence its reflection upon interchange of a and b intersects the a-axis at right angles at a = 2n, precisely where the parabola for n is reflected at the a-axis. (The same is of course true for a and b interchanged.)

text: Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings.[29][30]

text: Special cases[edit]

text: The Platonic sequence[edit]

text: The case n = 1 of the more general construction of Pythagorean triples has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows:

text:
Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to Pythagoras. (The latter) starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse.
...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which was obtained by the other method.

text: In equation form, this becomes:

text: a is odd (Pythagoras, c. 540 BC):

text: a is even (Plato, c. 380 BC):

text: It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (a, (a2 − 1)/2 and (a2 + 1)/2) by allowing a to take non-integer rational values. If a is replaced with the fraction m/n in the sequence, the result is equal to the 'standard' triple generator (2mn, m2 − n2,m2 + n2) after rescaling. It follows that every triple has a corresponding rational a value which can be used to generate a similar triangle (one with the same three angles and with sides in the same proportions as the original). For example, the Platonic equivalent of (56, 33, 65) is generated by a = m/n = 7/4 as (a, (a2 –1)/2, (a2+1)/2) = (56/32, 33/32, 65/32). The Platonic sequence itself can be derived[clarification needed] by following the steps for 'splitting the square' described in Diophantus II.VIII.

text: The Jacobi–Madden equation[edit]

text: The equation,

text: is equivalent to the special Pythagorean triple,

text: There is an infinite number of solutions to this equation as solving for the variables involves an elliptic curve. Small ones are,

text: Equal sums of two squares[edit]

text:
One way to generate solutions to

a

2

+

b

2

=

c

2

+

d

2

{\displaystyle a^{2}+b^{2}=c^{2}+d^{2}}

 is to parametrize a, b, c, d in terms of integers m, n, p, q as follows:[31]

text: Equal sums of two fourth powers[edit]

text: Given two sets of Pythagorean triples,

text: the problem of finding equal products of a non-hypotenuse side and the hypotenuse,

text: is easily seen to be equivalent to the equation,

text:
and was first solved by Euler as

a
,
b
,
c
,
d
=
133
,
59
,
158
,
134

{\displaystyle a,b,c,d=133,59,158,134}

. Since he showed this is a rational point in an elliptic curve, then there is an infinite number of solutions. In fact, he also found a 7th degree polynomial parameterization.

text: Descartes' Circle Theorem[edit]

text: For the case of Descartes' circle theorem where all variables are squares,

text: Euler showed this is equivalent to three simultaneous Pythagorean triples,

text:
There is also an infinite number of solutions, and for the special case when

a
+
b
=
c

{\displaystyle a+b=c}

, then the equation simplifies to,

text:
with small solutions as

a
,
b
,
c
,
d
=
3
,
5
,
8
,
14

{\displaystyle a,b,c,d=3,5,8,14}

 and can be solved as binary quadratic forms.

text: Almost-isosceles Pythagorean triples[edit]

text: No Pythagorean triples are isosceles, because the ratio of the hypotenuse to either other side is √2, but √2 cannot be expressed as the ratio of 2 integers.

text: There are, however, right-angled triangles with integral sides for which the lengths of the non-hypotenuse sides differ by one, such as,

text: and an infinite number of others. They can be completely parameterized as,

text:
where {x, y} are the solutions to the Pell equation

x

2

−
2

y

2

=
−
1

{\displaystyle x^{2}-2y^{2}=-1}

.

text: When it is the longer non-hypotenuse side and hypotenuse that differ by one, such as in

text: then the complete solution is

text: which also shows that all odd numbers (greater than 1) appear in a primitive Pythagorean triple.

text: Fibonacci numbers in Pythagorean triples[edit]

text: Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. The length of the longer leg of this triangle is equal to the sum of the three sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle.

text: Generalizations[edit]

text: There are several ways to generalize the concept of Pythagorean triples.

text: Pythagorean quadruple[edit]

text: A set of four positive integers a, b, c and d such that a2 + b2+ c2 = d2 is called a Pythagorean quadruple. The simplest example is (1, 2, 2, 3), since 12 + 22 + 22 = 32. The next simplest (primitive) example is (2, 3, 6, 7), since 22 + 32 + 62 = 72.

text: All quadruples are given by the formula

text: Pythagorean n-tuple[edit]

text: Using the simple algebraic identity,

text: for arbitrary x0, x1, it is easy to prove that the square of the sum of n squares is itself the sum of n squares by letting x0 = x22 + x32 + ... + xn2 and then distributing terms.[32] One can see how Pythagorean triples and quadruples are just the particular cases x0 = x22 and x0 = x22 + x32, respectively, and so on for other n, with quintuples given by

text: Since the sum F(k,m) of k consecutive squares beginning with m2 is given by the formula,[33]

text: one may find values (k, m) so that F(k,m) is a square, such as one by Hirschhorn where the number of terms is itself a square,[34]

text: and v ≥ 5 is any integer not divisible by 2 or 3. For the smallest case v = 5, hence k = 25, this yields the well-known cannonball-stacking problem of Lucas,

text: a fact which is connected to the Leech lattice.

text: In addition, if in a Pythagorean n-tuple (n ≥ 4) all addends are consecutive except one, one can use the equation,[35]

text:
Since the second power of p cancels out, this is only linear and easily solved for as

p
=

F
(
k
,
m
)
−
1

2

{\displaystyle p={\tfrac {F(k,m)-1}{2}}}

 though k, m should be chosen so that p is an integer, with a small example being k = 5, m = 1 yielding,

text: Thus, one way of generating Pythagorean n-tuples is by using, for various x,[36]

text: where q = n–2 and where

text: Fermat's Last Theorem[edit]

text: A generalization of the concept of Pythagorean triples is the search for triples of positive integers a, b, and c, such that an + bn = cn, for some n strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem because it took longer than any other conjecture by Fermat to be proven or disproven. The first proof was given by Andrew Wiles in 1994.

text: n − 1 or n nth powers summing to an nth power[edit]

text: Another generalization is searching for sequences of n + 1 positive integers for which the nth power of the last is the sum of the nth powers of the previous terms. The smallest sequences for known values of n are:

text:
n = 3: {3, 4, 5; 6}.
n = 4: {30, 120, 272, 315; 353}
n = 5: {19, 43, 46, 47, 67; 72}
n = 7: {127, 258, 266, 413, 430, 439, 525; 568}
n = 8: {90, 223, 478, 524, 748, 1088, 1190, 1324; 1409}

text:
For the n=3 case, in which

x

3

+

y

3

+

z

3

=

w

3

,

{\displaystyle x^{3}+y^{3}+z^{3}=w^{3},}

 called the Fermat cubic, a general formula exists giving all solutions.

text: A slightly different generalization allows the sum of (k + 1) nth powers to equal the sum of (n − k) nth powers. For example:

text: (n = 3): 13 + 123 = 93 + 103, made famous by Hardy's recollection of a conversation with Ramanujan about the number 1729 being the smallest number that can be expressed as a sum of two cubes in two distinct ways.

text: There can also exist n − 1 positive integers whose nth powers sum to an nth power (though, by Fermat's last theorem, not for n = 3); these are counterexamples to Euler's sum of powers conjecture. The smallest known counterexamples are[37][38][13]

text:
n = 4: (95800, 217519, 414560; 422481)
n = 5: (27, 84, 110, 133; 144)

text: Heronian triangle triples[edit]

text: A Heronian triangle is commonly defined as one with integer sides whose area is also an integer, and we shall consider Heronian triangles with distinct integer sides. The lengths of the sides of such a triangle form a Heronian triple (a, b, c) provided a < b < c. Clearly, any Pythagorean triple is a Heronian triple, since in a Pythagorean triple at least one of the legs a, b must be even, so that the area ab/2 is an integer. Not every Heronian triple is a Pythagorean triple, however, as the example (4, 13, 15) with area 24 shows.

text: If (a, b, c) is a Heronian triple, so is (ma, mb, mc) where m is any positive integer greater than one. The Heronian triple (a, b, c) is primitive provided a, b, c are pairwise relatively prime (as with a Pythagorean triple). Here are a few of the simplest primitive Heronian triples that are not Pythagorean triples:

text: By Heron's formula, the extra condition for a triple of positive integers (a, b, c) with a < b < c to be Heronian is that

text: or equivalently

text: be a nonzero perfect square divisible by 16.

text: Application to cryptography[edit]

text: Primitive Pythagorean triples have been used in cryptography as random sequences and for the generation of keys.[39]


document_title: Illegal entry

question_text: what is the charge for entering the u.s. illegally

text: Illegal entry

text: By country[edit]

text: India[edit]

text: Presently, India is constructing a fence along the border to restrict illegal traffic from Bangladesh.[7] The Indo-Bangladeshi barrier is 4,000 kilometers (2,500 mi) long.  The stated aim of the fence is to stop infiltration of terrorists, prevent smuggling, and to bring a close to illegal immigration from Bangladesh.[8][9]

text: Australia[edit]

text: In Australia, mandatory immigration detention was revived in 1992 for all foreigners who arrive in Australia without a visa. That only "border applicants" are subject to detention has sparked criticism, as it is claimed to unfairly discriminate against certain migrants.[10]

text: Suspected Irregular Entry Vessel was the name used by the Australian Defence Force or Australian Coastwatch for maritime vessels which were suspected to be attempting to reach Australia without authorisation. In practice, these boats were often carrying asylum seekers who had departed from Indonesia on the final leg of a journey which started in countries such as Iraq and Afghanistan after paying people smugglers.

text: Palestine[edit]

text: During 1933-1948, the British government limited Jewish immigration to Palestine with quotas, and following the rise of Nazism to power in Germany, illegal immigration to Palestine commenced. The illegal immigration was known as Aliyah Bet ("secondary immigration"), or Ha'apalah, and was organized by the Mossad Le'aliyah Bet, as well as by the Irgun. Immigration was done mainly by sea, and to a lesser extent overland through Iraq and Syria. Beginning in 1939 Jewish immigration was further restricted, limiting it to 75,000 individuals for a period of five years after which immigration was to end completely. During World War II and the years that followed until independence, Aliyah Bet became the main form of Jewish immigration to Palestine. Despite British efforts to curb the illegal immigration, during the 14 years of its operation, 110,000 Jews immigrated to Palestine.

text: Turkey[edit]

text: Turkey, which is a transit point for unauthorized migrants trying to reach Europe, has been accused of being unable to secure its borders with Greece. Since 1996, 40 unauthorized migrants have been killed by mines, after entering Greek territory in Evros.[11] In 2001, about 800 illegal immigrants were rescued by the Greek coastguards after a fire broke out on board the Turkish-flagged Brelner, believed to have set sail from the Turkish port of İzmir, probably en route to Italy.[12]  Once in July 2004 and a second time in May 2006, Hellenic Coast Guard ships were caught on film cruising as near as a few hundred meters off the Turkish coast and abandoning clandestines to the sea. This practice resulted in the drowning of six people between Chios and Karaburun on 26 September 2006 while three others disappeared and 31 were saved by Turkish gendarmes and fishermen.  Three of the drowned were Tunisians, one was Algerian, one Palestinian and the other Iraqi. The three disappeared were also Tunisians.[13][14][15]

text: As a result of bilateral negotiations, a readmission agreement was signed between Turkey and Greece in November 2001 and went into effect in April 2002. For third country nationals, this protocol gives the parties 14 days to inform each other of the number of persons to be returned after the date of illegal entry. For nationals of the two countries the authorities can make use of simplified procedures. But the strict application of the agreement is reported to have retrograded as of 2003.

text: China[edit]

text: People's Republic of China is building a security barrier along its border with North Korea to prevent defectors or refugees from North Korea from entering the country illegally.[16]

text: On May 13, 2004, the People's Republic of China sentenced Yang Jianli, a Chinese dissident with U.S. residency, to five years in prison for espionage and illegal entry.

text: Hong Kong[edit]

text: The illegal entry of Vietnamese refugees was a concern for the Hong Kong government for 25 years. The issue was first resolved in 2000. Between 1975 and 1999, 143,700 Vietnamese refugees were resettled in other countries and more than 67,000 Vietnamese migrants were repatriated.[17]

text: Iran[edit]

text: On 23 March 2007, 15 British Royal Navy personnel, from HMS Cornwall, were surrounded by the Navy of the Iranian Revolutionary Guards and were subsequently detained off the Iraq-Iran coast. Iran's director general for Western European affairs, Ibrahim Rahimpour, said that the British boats had made "illegal entry" into Iranian territorial waters and that the personnel "were arrested by border guards for investigation and questioning".[18] On 24 March, the Iranian Fars News Agency said the navigational equipment seized on the British boats shows the sailors were aware that they were operating in Iranian waters. On the same day, General Ali Reza Afshar, a top military official, said the sailors had confessed to illegal entry into Iran's waters.[19]

text: Egypt[edit]

text: On January 25, 2008, Egyptian security forces blocked almost all illegal entry points along the border with Gaza to try to stem the flow of Palestinians wanting to leave.[20] Egyptian forces in riot gear erected barbed wire and chain-link fences along the border to prevent more Palestinians from crossing.[21]

text: Thailand[edit]

text: On June 5, 2006, 231 Hmong refugees fleeing Laos were detained by police in Amphoe Khao Kho, Phetchabun Province for illegal entry into Thailand. The Hmong were seeking asylum at the province's Ban Huay Nam Khao, where about 6,500 ethnic Hmong are being sheltered, but the Thai military refused to allow them to stay. Most of the Hmong claim they fought against the communists in the Secret War.[22]

text: Malaysia[edit]

text: Most unlawful migrants to Malaysia are from nearby Southeast Asian countries such as Indonesia particularly on Indonesia–Malaysia border via maritime boundary of Strait of Malacca and land border on the island of Borneo. The land border on Malaysia–Thailand mostly used by Bangladeshi & Rakhine people of Myanmar. The Malaysia–Philippines border normally used by Mindanao people via Celebes Sea.[citation needed]

text: United States[edit]

text: There are an estimated half a million illegal entries into the United States each year.[23] Six to seven million people reside in the United States after having evaded the Immigration Inspectors or Border Patrol.[24]

text: The illegal entry of non-nationals into the United States is a misdemeanor according to the Immigration and Nationality Act of 1965, which prohibits non-nationals from entering or attempting to enter the United States at any time or place which has not been designated by an immigration officer, and also prohibits non-nationals from eluding inspection by immigration officers.[25]

text: The U.S. Customs and Border Protection is responsible for apprehending individuals attempting to enter the United States illegally. The United States Border Patrol is its mobile uniformed law enforcement arm, responsible for deterrence, detection and apprehension of those who enter the United States without authorization from the government or outside the designated ports of entry.

text:
The unfenced, rural mountainous and desert border between Arizona and Mexico has become a major entrance area for unlawful migration to the United States, due in part to the increased difficulty of crossing illegally into California. Each year
there are several hundred migrant deaths along the Mexico-U.S. border.[26] Often, migrants employ people smugglers (known as "coyotes") who promise a safe passage into the United States and are paid up to 3 to 4 thousand[clarification needed] for passages that include long walking distances or 7 to 14 thousand[clarification needed] for an easier travel per person they assist in crossing the border.[27] The difficulty and expense of the journey has prompted many migrant workers to stay in the United States longer or indefinitely.[28]

text:
According to the U.S. Border Patrol, there were 1,954 migrant deaths along the Mexico-U.S. border between 1998 and 2004.[29] In 1993, 283 Chinese migrants attempted illegal entry into the United
States via a sea vessel; ten of them arrived dead. Reference: Bill Ward[30][31]

text: United Kingdom[edit]

text: A recent study into irregular immigration to the United Kingdom states that "most irregular migrants have committed administrative offences rather than a serious crime".[32] In 2004, illegal entry action was initiated against 36,550 migrants; it was estimated that more migrants overstay their visa than enter clandestinely, hence considered unlikely that there are large numbers of people crossing UK borders without permission.[32]

text: Many of the routes for clandestine entry are very dangerous. There were 72 documented deaths that are thought to have occurred while entering UK between 1993 and 2002.[32] The most common cause of death while entering the UK (61 of the deaths recorded) was suffocation in a lorry while being smuggled into the UK by ferry; Others drowned in the English Channel, froze to death as stowaways on an aeroplane and were crushed by trains in the Channel Tunnel.[32]


document_title: Operation Sovereign Borders

question_text: when did the stop the boats campaign start

text: Operation Sovereign Borders

text: Background[edit]

text: During the 2013 federal election, the Abbott-led Coalition campaigned on a policy that, if elected to government, they would "stop the boats" and would launch Operation Sovereign Borders, combining the resources of multiple government bodies under direct control of a three star general. Following the election, Angus Campbell was promoted and appointed to oversee the operations.[3]

text: Following the 2013 election, the portfolio of the Minister for Immigration was renamed as the Minister for Immigration and Border Protection. The appointed ministers, initially Scott Morrison and subsequently Peter Dutton, refused to release information on asylum seeker boat arrivals as they occurred,[4] and a weekly media briefing was announced.[5][6] In January 2014, having not held a media briefing for almost a month, Morrison announced that briefings would be held on what he described as "an as needs basis".[7] On 10 July 2014, Morrison stated that the secrecy policy was put in place by Lieutenant General Campbell, which had been rigorously implemented by ministers, their advisers, and various government departments.[8]

text: Policy proposals[edit]

text: Regional Deterrence Framework[edit]

text: On 23 August 2013, during the election campaign, the Coalition announced a key component of Operation Sovereign Borders called the Regional Deterrence Framework.[9] Budgeted at A$420 million, the RDF aimed to engage with other countries in the region, particularly Indonesia, to prevent asylum seeker vessels leaving for Australia. The framework included a $20 million proposal (titled "The Indonesian community engagement programme") which was to include:[10]

text:
communications campaigns to raise awareness within local villages that people smuggling is a criminal activity;
a capped boat buy-back scheme that was to provide an incentive for owners of decrepit and dangerously unsafe boats to sell their boats to government officials rather than people smugglers;
support for wardens in local communities, whose role was to be to provide intelligence information to the Indonesian National Police on people smuggling operations;
the option in exceptional circumstances for bounty payments for the provision of information resulting in significant disruptions or arrests leading to convictions.

text: The "buy-the-boats" plan was widely ridiculed,[11] with fact-checking group PolitiFact Australia[12] calling the proposal "ridiculous".[13] Lieutenant General Campbell told a Senate Estimates committee that, two months into the OSB program, no boats had been purchased because Indonesia did not support the idea, although he stated that the measure remained available.[14]

text: Communication campaign[edit]

text: The government runs a "communication campaign to counter people smuggling"[15] with advertisements in multiple languages,[16] targeting "press, radio, social and search media" across Australia. Between January and May 2015, $750,000 had been spent on the campaign.[17]

text: Structure[edit]

text: Operation Sovereign Borders operates as a Joint Agency Taskforce (JATF), with the support of a range of government agencies, organised as three operational task groups:[18]

text:
Detection, Interception and Transfer Task Group – led by the Australian Border Force (incorporating Maritime Border Command)
Disruption and Deterrence Task Group – led by the Australian Federal Police
Offshore Detention and Returns Task Group – led by the Australian Border Force

text: Commanders[edit]

text:
Rank
Name
Post-nominals
Service
Term began
Term ended

Lieutenant General
Angus Campbell
DSC, AM
Army
18 September 2013
16 May 2015

Major General
Andrew Bottrell
CSC & Bar, DSM
Army
16 May 2015
1 February 2017

Air Vice Marshal
Stephen Osborne
AM, CSC
RAAF
1 February 2017
Incumbent

text: Outcomes[edit]

text: Abbott's government claimed a ninety per cent reduction in illegal maritime arrivals.[19] There were 207 in November 2013, as opposed to 2,629 in November 2012.[20][21][22]

text: In response, Shadow Minister for Immigration and Border Protection Richard Marles claimed there was a 40 per cent reduction in arrivals in the month following the introduction of the Regional Resettlement Arrangement with Papua New Guinea, shortly before the 2013 election.

text: On 30 March 2014, the Coalition announced that 100 days had passed since a "successful boat arrival".[23] On 19 June, the Government announced six months since the last successful illegal boat arrival.[24]

text: On 7 July 2014, a vessel containing 157 (mostly Tamil) asylum seekers from India was intercepted by Australian authorities 27 kilometres (15 nmi) from Christmas Island. The government refused to confirm the existence, location, or status of the boat, until the High Court placed an injunction on any attempted refoulement of the vessel's passengers to Sri Lanka. Pre-empting the decision of the court, Minister for Immigration and Border Protection Morrison announced that the people on the boat would be transferred to the Curtin Immigration Reception and Processing Centre in Western Australia, where they would be assessed by Indian consular officials under an arrangement made with that country to repatriate any Indian citizens or residents.[25] On 2 August, Morrison announced that the group had refused to meet with Indian officials and were then transferred to the Nauru Regional Processing Centre.[26]

text: On 6 August 2015, the new immigration minister Peter Dutton announced it had been 12 months since the last successful people smuggling operation, with the last SIEV arriving in Australia's care in July 2014. The ABC News' Fact Check subsequently listed the Coalition's "We Will Stop the Boats" promise as delivered.[27]

text:
Unlawful arrivals by boat per month

Month
Boats
Persons
Notes

From 18 September 2013
5
205

October 2013
5
339

November 2013
5
207

December 2013
7
355

January 2014
0
1
[28]

July 2014
1
157
[29]

Source: "Operation Sovereign Borders:log of boat arrivals and other asylum seeker incidents". ABC News. Australia. Retrieved 5 February 2015. Months with no arrivals are not listed.

text: Turnback operations[edit]

text: The number of arrivals given in OSB operational updates is defined as those "transferred to Australian immigration authorities",[30] and does not include arrivals in Australian territorial waters who have been subject to a turnback operation—that is, sent out of Australian waters on their own vessel, or an Australian vessel employed for this purpose.[31] As of 7 February 2014, The Australian newspaper estimated that at least "six boatloads" of asylum seekers had been subject to turnbacks by OSB authorities.[32]

text: On 15 January 2014, an orange fibreglass "survival capsule", containing about 60 asylum seekers, came ashore at Cikepuh in West Java. A second containing 34 people arrived at Pangandaran on 5 February.[33] The Daily Telegraph reported that the Australian government was believed to have purchased eleven of the capsules from Singapore at a cost of around $500,000.[34]

text: In May 2014, Australia was alleged to have placed two persons who had arrived earlier in the year onto a boat with other asylum seekers which was turned back to Indonesia.[35]

text: In January 2015, Minister Dutton announced that 15 vessels, containing 429 asylum seekers in total, had been subject to turnback operations of some kind towards Indonesia or Sri Lanka since the beginning of OSB.[36]

text: In July 2015, Labor Shadow Minister Richard Marles conceded that "Offshore processing and regional resettlement, together with the Coalition's policy of turn-backs, is what actually stopped the boats."[37]

text: In August 2015, Dutton stated that, since December 2013, 633 people on 20 vessels have been subject to turnback operations, including a boat from Vietnam in July.[38] In March 2016, Dutton stated that 698 people on 25 vessels had been turned back since the beginning of the OSB program.[39]

text: Resettlement[edit]

text: In 2014, the status of refugees sent to offshore processing centres in Nauru and Papua New Guinea was decided: 13 people (9 people from Iran and 4 people from Pakistan) were granted asylum, while 7 people (from Iran, Pakistan, and Cameroon) received negative assessments. The asylum protection in Nauru was valid from 2014 for up to 5 years.[40] As of 2015, more than 400 people who had their refugee claims rejected had been returned home from the Australian-run detention centre in Papua New Guinea, some of which did so voluntarily.[41]

text: Response[edit]

text: International response[edit]

text: The Indonesian government has voiced concern over the operation due to its implications for Indonesia's national sovereignty.[42][43] A member of the Golkar party, Tantowi Yahya, described the plan as "offensive", and officials from the Indonesian Navy said "forcing the boats back would also unfairly shift the burden of dealing with the asylum-seeker problem back on Indonesia".[44] The policy also came under fire from refugee advocates.[45]

text: On 26 September 2013, Indonesian Foreign Minister Marty Natalegawa took the "unusual step" of releasing details of his talks about the policy with his Australian counterpart Julie Bishop,[46] which was later blamed on a clerical error.[47]

text: Australia has apologised for violating Indonesian waters during their "tow back" operations.[48] These incursions occurred after Chief of the Defence Force David Hurley stripped naval personnel of workplace safety protections that would have required them to exercise "reasonable care" to protect their safety and that of the refugees.[49] On 21 January 2014, Customs (now Australian Border Force) and Defence announced that a joint review would be conducted to investigate the circumstances under which Australian naval vessels entered Indonesian territorial waters.[50] The inquiry, which covered the period between 1 December 2013 and 20 January 2014, found that two Royal Australian Navy frigates had crossed into Indonesian territory four times during the period, while Customs vessels did so on another two occasions.[51] In response, one Australian Navy officer lost his command, while several others were disciplined.[52] Indonesia has responded to the incursions by deploying military assets to intercept people-smuggling boats.[53][54]

text: Media response[edit]

text: Several journalists and media outlets have expressed concern and frustration over the tightly-controlled release of information about Operation Sovereign Borders, usually restricted to the weekly briefings held on Friday afternoons. In the weekly briefings, both Minister Morrison and Lieutenant General Campbell have refused to discuss "operational" or "on-water" matters in response to questions from journalists. The Minister has rationalised the control of information by stating that the government was not "operating a shipping news service for people smugglers".[55]

text: On 22 January 2014, the ABC broadcast allegations that Royal Australian Navy personnel had mistreated asylum seekers during an OSB operation, including video footage of passengers receiving medical treatment in Indonesia for burns on their hands, which they claimed were sustained when they were forced to touch a hot boat engine.[56] Minister Morrison downplayed refugee claims of being abused by the navy, saying that they had "strong motivations to fabricate stories", and was among several ministers to call for the ABC to apologise to the Navy.[57] Foreign Minister Julie Bishop said: "if there is any co-operation we can extend to ensure that these allegations are scotched then we'd be prepared to do it".[58] The ABC's Media Watch program opined that ABC News had "over-reached" when reporting the story, and should have been more thorough in verifying the claims.[59] On 4 February, ABC managing director Mark Scott issued a statement in which he said that "The wording around the ABC's initial reporting needed to be more precise on that point", referring to the video footage verifying the injuries but not how they had occurred.[60] On 7 February, Fairfax Media's correspondent in Indonesia, Michael Bachelard, interviewed Yousif Ibrahim Fasher who repeated the initial allegations, as well as several further claims of mistreatment and possible breaches of maritime law.[61]

text: Legal response[edit]

text: 53 Australian legal scholars signed a joint statement condemning the transfer of asylum seekers to Sri Lanka, claiming a breach of the Convention Relating to the Status of Refugees, the Universal Declaration of Human Rights, and the International Covenant on Civil and Political Rights.[62][63]

text: On 7 July 2014, the High Court of Australia placed an injunction on the transfer of 153 asylum seekers to Sri Lankan authorities while the full bench of the court considered a challenge to the handover mounted by refugee advocates on the grounds that the government was breaching "non-refoulement" obligations under international law.[64] Under Article 33 of the Convention Relating to the Status of Refugees, in which Australia is a signatory, this principle forbids a nation state from sending a refugee back to anywhere where they may face persecution.[65]

text: In 2016, Operation Sovereign Borders is still in effect, with international law allowing Australia to deny any vessel entry into their territorial waters, as long as Australia does not return these asylum seekers to a place in which they may be threatened.[66] Australia therefore recognises there to be no violation of their international obligations, as refusing entry of asylum seekers does not equate to a breach of non-refoulement.