Congruent Number

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## Congruent number problem

## Relation to elliptic curves

## Current progress

## Notes

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Congruent Number

In mathematics, a **congruent number** is a positive integer that is the area of a right triangle with three rational number sides.^{[1]} A more general definition includes all positive rational numbers with this property.^{[2]}

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, ... (sequence in the OEIS)

Congruent number table: n ≤ 120 (sequence in the OEIS)

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 is not a congruent number.

If q is a congruent number then *s*^{2}*q* 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

- .

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.

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.^{[3]} Every congruum is a congruent number, and every congruent number is a product of a congruum and the square of a rational number.^{[4]} 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.^{[5]}

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.^{[2]} 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:

Then set *x* = *n*(*a*+*c*)/*b* and
*y* = 2*n*^{2}(*a*+*c*)/*b*^{2}.
A calculation shows

and y is not 0 (if *y* = 0 then *a* = -*c*, so *b* = 0, but ()*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* = (*x*^{2} - *n*^{2})/*y*,
*b* = 2*nx*/*y*, and *c* = (*x*^{2} + *n*^{2})/*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 *y*^{2} = *x*^{3} - *xn*^{2} = *x*(*x*^{2} - *n*^{2})
that if x and y are positive then *x*^{2} - *n*^{2} 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
*y*^{2} = *x*^{3} - *n*^{2}*x* has a rational point with y not equal to 0.
It can be shown (as a nice 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

where

Much work has been done classifying congruent numbers.

For example, it is known^{[6]} 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^{[7]} 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.

**^**Weisstein, Eric W. "Congruent Number".*MathWorld*.- ^
^{a}^{b}Koblitz, Neal (1993),*Introduction to Elliptic Curves and Modular Forms*, New York: Springer-Verlag, p. 3, ISBN 0-387-97966-2 **^**Ore, Øystein (2012),*Number Theory and Its History*, Courier Dover Corporation, pp. 202-203, ISBN 978-0-486-13643-1.**^**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.**^**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.**^**Paul Monsky (1990), "Mock Heegner Points and Congruent Numbers",*Mathematische Zeitschrift*,**204**(1): 45-67, doi:10.1007/BF02570859**^**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.

- Alter, Ronald (1980), "The Congruent Number Problem",
*American Mathematical Monthly*, Mathematical Association of America,**87**(1): 43-45, doi:10.2307/2320381, JSTOR 2320381 - Chandrasekar, V. (1998), "The Congruent Number Problem" (PDF),
*Resonance*,**3**(8): 33-45, doi:10.1007/BF02837344 - Dickson, Leonard Eugene (2005), "Chapter XVI",
*History of the Theory of Numbers*, Dover Books on Mathematics, Volume II: Diophantine Analysis, Dover Publications, ISBN 978-0-486-44233-4 - see, for a history of the problem. - Guy, Richard (2004),
*Unsolved Problems in Number Theory*, Problem Books in Mathematics (Book 1) (3rd ed.), Springer, ISBN 978-0-387-20860-2, Zbl 1058.11001 - Many references are given it in. - Tunnell, Jerrold B. (1983), "A classical Diophantine problem and modular forms of weight 3/2",
*Inventiones Mathematicae*,**72**(2): 323-334, Bibcode:1983InMat..72..323T, doi:10.1007/BF01389327

- Weisstein, Eric W. "Congruent Number".
*MathWorld*. - 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).

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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