# Congruent number

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.[1][2] A more general definition includes all positive rational numbers with this property.[3]

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, ... (sequence A003273 in the OEIS)
Congruent number table: n ≤ 120
Congruent number table: n ≤ 120
—: non-Congruent number
C: square-free Congruent number
S: Congruent number with square factor
n 1 2 3 4 5 6 7 8
C C C
n 9 10 11 12 13 14 15 16
C C C
n 17 18 19 20 21 22 23 24
S C C C S
n 25 26 27 28 29 30 31 32
S C C C
n 33 34 35 36 37 38 39 40
C C C C
n 41 42 43 44 45 46 47 48
C S C C
n 49 50 51 52 53 54 55 56
S C S C S
n 57 58 59 60 61 62 63 64
S C C S
n 65 66 67 68 69 70 71 72
C C C C
n 73 74 75 76 77 78 79 80
C C C S
n 81 82 83 84 85 86 87 88
S C C C S
n 89 90 91 92 93 94 95 96
S C C C S
n 97 98 99 100 101 102 103 104
C C C
n 105 106 107 108 109 110 111 112
C C C S
n 113 114 115 116 117 118 119 120
S S C C 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.

If q is a congruent number then s2q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s), and vice versa. This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group

${\displaystyle \mathbb {Q} ^{*}/\mathbb {Q} ^{*2}}$,

where ${\displaystyle \mathbb {Q} ^{*}}$ 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.

## 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 (as of 2019) 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.[4] Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number.[5] 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.[6]

## Solutions

n is a congruent number if and only if the system

${\displaystyle x^{2}-ny^{2}=u^{2}}$, ${\displaystyle x^{2}+ny^{2}=v^{2}}$

has a solution where ${\displaystyle x,y,u}$, and ${\displaystyle v}$ are integers.[7]

Given a solution, the three numbers ${\displaystyle u^{2}}$, ${\displaystyle x^{2}}$, and ${\displaystyle v^{2}}$ will be in an arithmetic progression with common difference ${\displaystyle ny^{2}}$.

Furthermore, if there is one solution (where the right-hand sides are squares), then there are infinitely many: given any solution ${\displaystyle (x,y)}$, another solution ${\displaystyle (x',y')}$ can be computed from[8]

${\displaystyle x'=(xu)^{2}+n(yv)^{2}}$,
${\displaystyle y'=2xyuv}$.

For example, with ${\displaystyle n=6}$, the equations are:

${\displaystyle x^{2}-6y^{2}=u^{2}}$,
${\displaystyle x^{2}+6y^{2}=v^{2}}$.

One solution is ${\displaystyle x=5,y=2}$ (so that ${\displaystyle u=1,v=7}$). Another solution is

${\displaystyle x'=(5\cdot 1)^{2}+6(2\cdot 7)^{2}=1201}$,
${\displaystyle y'=2\cdot 5\cdot 2\cdot 1\cdot 7=140}$.

With this new ${\displaystyle x}$ and ${\displaystyle y}$, the right-hand sides are still both squares:

${\displaystyle u^{2}=1201^{2}-6\cdot 140^{2}=1324801=1151^{2}}$
${\displaystyle v^{2}=1201^{2}+6\cdot 140^{2}=1560001=1249^{2}}$.

Given ${\displaystyle x,y,u}$, and ${\displaystyle v}$, one can obtain ${\displaystyle a,b}$, and ${\displaystyle c}$ such that

${\displaystyle a^{2}+b^{2}=c^{2}}$, and ${\displaystyle {\frac {ab}{2}}=n}$

from

${\displaystyle a={\frac {v-u}{y}}}$, ${\displaystyle b={\frac {v+u}{y}}}$, ${\displaystyle c={\frac {2x}{y}}}$.

Then ${\displaystyle a,b}$ and ${\displaystyle c}$ are the legs and hypotenuse of a right triangle with area ${\displaystyle n}$.

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

## Relation to elliptic curves

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.[3] An alternative approach to the idea is presented below (as can essentially also be found in the introduction to Tunnell's paper).

Suppose a, b, c are numbers (not necessarily positive or rational) which satisfy the following two equations:

${\displaystyle {\begin{matrix}a^{2}+b^{2}&=&c^{2},\\{\tfrac {1}{2}}ab&=&n.\end{matrix}}}$

Then set x = n(a+c)/b and y = 2n2(a+c)/b2. A calculation shows

${\displaystyle y^{2}=x^{3}-n^{2}x}$

and y is not 0 (if y = 0 then a = -c, so b = 0, but (12)ab = n is nonzero, a contradiction).

Conversely, if x and y are numbers which satisfy the above equation and y is not 0, set a = (x2 - n2)/y, b = 2nx/y, and c = (x2 + n2)/y. A calculation shows these three numbers satisfy the two equations for a, b, and c above.

These two correspondences between (a,b,c) and (x,y) are inverses of each other, so we have a one-to-one correspondence between any solution of the two equations in a, b, and c and any solution of the equation in x and y with y nonzero. In particular, from the formulas in the two correspondences, for rational n we see that a, b, and c are rational if and only if the corresponding x and y are rational, and vice versa. (We also have that a, b, and c are all positive if and only if x and y are all positive; from the equation y2 = x3 - xn2 = x(x2 - n2) we see that if x and y are positive then x2 - n2 must be positive, so the formula for a above is positive.)

Thus a positive rational number n is congruent if and only if the equation y2 = x3 - n2x has a rational point with y not equal to 0. It can be shown (as an application of Dirichlet's theorem on primes in arithmetic progression) that the only torsion points on this elliptic curve are those with y equal to 0, hence the existence of a rational point with y nonzero is equivalent to saying the elliptic curve has positive rank.

Another approach to solving is to start with integer value of n denoted as N and solve

${\displaystyle N^{2}=ed^{2}+e^{2}}$

where

${\displaystyle {\begin{matrix}c&=&n^{2}/e+e\\a&=&2n\\b&=&n^{2}/e-e\end{matrix}}}$

## Smallest solutions

David Goldberg has computed congruent square-free numbers less than 104, along with the corresponding a and b values.[9]

## Current progress

Much work has been done classifying congruent numbers.

For example, it is known[10] that for a prime number p, the following holds:

• if p ≡ 3 (mod 8), then p is not a congruent number, but 2p is a congruent number.
• if p ≡ 5 (mod 8), then p is a congruent number.
• if p ≡ 7 (mod 8), then p and 2p are congruent numbers.

It is also known[11] that in each of the congruence classes 5, 6, 7 (mod 8), for any given k there are infinitely many square-free congruent numbers with k prime factors.

## Notes

1. ^ Weisstein, Eric W. "Congruent Number". MathWorld.
2. ^ Guy, Richard K. (2004). Unsolved problems in number theory ([3rd ed.] ed.). New York: Springer. pp. 195–197. ISBN 0-387-20860-7. OCLC 54611248.
3. ^ a b Koblitz, Neal (1993), Introduction to Elliptic Curves and Modular Forms, New York: Springer-Verlag, p. 3, ISBN 0-387-97966-2
4. ^ Ore, Øystein (2012), Number Theory and Its History, Courier Dover Corporation, pp. 202–203, ISBN 978-0-486-13643-1.
5. ^ Conrad, Keith (Fall 2008), "The congruent number problem" (PDF), Harvard College Mathematical Review, 2 (2): 58–73, archived from the original (PDF) on 2013-01-20.
6. ^ Darling, David (2004), The Universal Book of Mathematics: From Abracadabra to Zeno's Paradoxes, John Wiley & Sons, p. 77, ISBN 978-0-471-66700-1.
7. ^ Uspensky, J. V.; Heaslet, M. A. (1939). Elementary Number Theory. Vol. 2. McGraw Hill. p. 419.
8. ^ Dickson, Leonard Eugene (1966). History of the Theory of Numbers. Vol. 2. Chelsea. pp. 468–469.
9. ^ Goldberg, David (7 June 2021). "Triangle Sides for Congruent Numbers less than 10,000". arXiv:2106.07373 [math.NT].
10. ^ Paul Monsky (1990), "Mock Heegner Points and Congruent Numbers", Mathematische Zeitschrift, 204 (1): 45–67, doi:10.1007/BF02570859
11. ^ Tian, Ye (2014), "Congruent numbers and Heegner points", Cambridge Journal of Mathematics, 2 (1): 117–161, arXiv:1210.8231, doi:10.4310/CJM.2014.v2.n1.a4, MR 3272014.