Congruent number

thumb|Right triangle with the area 6, a congruent number.

In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.

The sequence of (integer) congruent numbers starts with

:5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, ...

{| class="wikitable floatright" style="text-align:center"

|+ Congruent number table: ≤ 120 —: non-Congruent number C: square-free Congruent number S: Congruent number with square factor

! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8

| — || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16

| — || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 17 !! 18 !! 19 !! 20 !! 21 !! 22 !! 23 !! 24

| — || — || — || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S

! 25 !! 26 !! 27 !! 28 !! 29 !! 30 !! 31 !! 32

| — || — || — || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 33 !! 34 !! 35 !! 36 !! 37 !! 38 !! 39 !! 40

| — || style="background-color:#FFC0CB" | C || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 41 !! 42 !! 43 !! 44 !! 45 !! 46 !! 47 !! 48

| style="background-color:#FFC0CB" | C || — || — || — || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 49 !! 50 !! 51 !! 52 !! 53 !! 54 !! 55 !! 56

| — || — || — || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S

! 57 !! 58 !! 59 !! 60 !! 61 !! 62 !! 63 !! 64

| — || — || — || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S || —

! 65 !! 66 !! 67 !! 68 !! 69 !! 70 !! 71 !! 72

| style="background-color:#FFC0CB" | C || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 73 !! 74 !! 75 !! 76 !! 77 !! 78 !! 79 !! 80

| — || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S

! 81 !! 82 !! 83 !! 84 !! 85 !! 86 !! 87 !! 88

! 89 !! 90 !! 91 !! 92 !! 93 !! 94 !! 95 !! 96

! 97 !! 98 !! 99 !! 100 !! 101 !! 102 !! 103 !! 104

| — || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || —

! 105 !! 106 !! 107 !! 108 !! 109 !! 110 !! 111 !! 112

| — || — || — || — || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S

! 113 !! 114 !! 115 !! 116 !! 117 !! 118 !! 119 !! 120

| — || — || — || style="background-color:#98FB98" | S || style="background-color:#98FB98" | S || style="background-color:#FFC0CB" | C || style="background-color:#FFC0CB" | C || style="background-color:#98FB98" | S

For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers. The triangle sides demonstrating a number is congruent can have very large numerators and denominators, for example 263 is the area of a triangle whose two shortest sides are 16277526249841969031325182370950195/2303229894605810399672144140263708 and 4606459789211620799344288280527416/61891734790273646506939856923765.

If is a congruent number then is also a congruent number for any natural number (just by multiplying each side of the triangle by ), and vice versa. This leads to the observation that whether a nonzero rational number is a congruent number depends only on its residue in the group

:<math>\mathbb{Q}^{*}/\mathbb{Q}^{*2}, </math>

where <math>\mathbb{Q}^{*}</math> is the set of nonzero rational numbers.

Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers when speaking about congruent numbers.

History

The congruent number problem was first formulated by the 10th–century Persian mathematician Al-Khazin. Congruent numbers were first tabulated in Arabic manuscripts by the 10th century, with 5 and 6 among the earliest known examples. In the 13th century, Fibonacci identified 7 as a congruent number and noted that 1 is not. The first accepted proof of the non-congruence of 1 was later given by Pierre de Fermat, who also proved that 2 and 3 are not congruent numbers.

According to Arnoud Vrolijk and Jan Hogendijk, despite speculation regarding an Indian origin for the congruent number problem, standard historical consensus provides no evidence to support this.

Congruent number problem

The question of determining whether a given rational number is a congruent number is called the congruent number problem. , this problem has not been brought to a successful resolution. Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Fermat's right triangle theorem, named after Pierre de Fermat, states that no square number can be a congruent number. However, in the form that every congruum (the difference between consecutive elements in an arithmetic progression of three squares) is non-square, it was already known (without proof) to Fibonacci. Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number. However, determining whether a number is a congruum is much easier than determining whether it is congruent, because there is a parameterized formula for congrua for which only finitely many parameter values need to be tested.

Solutions

n is a congruent number if and only if the system

:<math>x^2 - n y^2 = u^2</math>, <math>x^2 + n y^2 = v^2</math>

has a solution where <math>x, y, u</math>, and <math>v</math> are integers.

Given a solution, the three numbers <math>u^2</math>, <math>x^2</math>, and <math>v^2</math> will be in an arithmetic progression with common difference <math>n y^2</math>.

Furthermore, if there is one solution (where the right-hand sides are squares), then there are infinitely many: given any solution <math>(x, y)</math>,

another solution <math>(x', y')</math> can be computed from

:<math>x' = (x u)^2 + n (y v)^2, </math>

:<math>y' = 2 x y u v.</math>

For example, with <math>n = 6</math>, the equations are:

:<math>x^2 - 6 y^2 = u^2, </math>

:<math>x^2 + 6 y^2 = v^2. </math>

One solution is <math>x = 5, y = 2</math> (so that <math>u = 1, v = 7</math>). Another solution is

:<math>x' = (5 \cdot 1)^2 + 6 (2 \cdot 7)^2 = 1201, </math>

:<math>y' = 2 \cdot 5 \cdot 2 \cdot 1 \cdot 7 = 140. </math>

With this new <math>x'</math> and <math>y'</math>, the new right-hand sides are still both squares:

:<math>u'^2 = 1201^2 - 6 \cdot 140^2 = 1324801 = 1151^2, </math>

:<math>v'^2 = 1201^2 + 6 \cdot 140^2 = 1560001 = 1249^2. </math>

Using <math>x'=1201, y'=140, u', v'</math> as above gives

:<math>u=1,727,438,169,601</math>

:<math>v=2,405,943,600,001</math>

Given <math>x, y, u</math>, and <math>v</math>, one can obtain <math>a, b</math>, and <math>c</math> such that

:<math>a^2 + b^2 = c^2</math>, and <math>\frac{ab}{2} = n</math>

from

:<math>a = \frac{v - u}{y}, \quad b = \frac{v + u}{y}, \quad c = \frac{2x}{y}.</math>

Then <math>a, b</math> and <math>c</math> are the legs and hypotenuse of a right triangle with area <math>n</math>.

The above values <math>(x, y, u, v) = (5, 2, 1, 7)</math> produce <math>(a, b, c) = (3, 4, 5)</math>. The values <math>(1201, 140, 1151, 1249)</math> give <math>(a, b, c) = (7/10, 120/7, 1201/70)</math>. Both of these right triangles have area <math>n = 6</math>.

Relation to elliptic curves

According to Leonard Eugene Dickson's compilation of historical texts, a tenth-century Arabic manuscript by Mohammed Ben Alhocain states that the core goal of the theory of rational right triangles is to identify a square that, when increased or decreased by a given number, results in another square. In modern terminology, this task translates to locating a rational point of infinite order on the elliptic curve <math>my^2 = x^3 - x</math>.

The question of whether a given number is congruent turns out to be equivalent to the condition that a certain elliptic curve has positive rank.

if , then is not a congruent number, but 2 is a congruent number.
if , then is a congruent number.
if , then and 2 are congruent numbers.

It is also known that in each of the congruence classes , for any given there are infinitely many square-free congruent numbers with prime factors.

Notes

References

– see, for a history of the problem.
– Many references are given in it.

External links

A short discussion of the current state of the problem with many references can be found in Alice Silverberg's Open Questions in Arithmetic Algebraic Geometry (Postscript).
A Trillion Triangles - mathematicians have resolved the first one trillion cases (conditional on the Birch and Swinnerton-Dyer conjecture).