congruent numbers

A number is called congruent it is the area of a right triangle with rational sides.

For example (see Figure aside), 7 is congruent because

$\left(\frac{35}{12}\right)^2 +\left(\frac{24}{5}\right)^2=\left(\frac{337}{60}\right)^2\,,$

and $\frac{1}{2}\cdot\frac{35}{12}\cdot\frac{24}{5}=7$ .

Equivalently, a number $n$ is congruent if there exist 3 rational squares $a^2$ , $b^2$ , and $c^2$ in arithmetic progression such that $b^2-a^2=c^2-b^2=n$ .

The problem of determining if a number is congruent is old and difficult. The numbers involved are often very large. For example, according to Zeigel, 157 is proved congruent by the right triangle with legs $\frac{411340519227716149383203}{21666555693714761309610}$ and $\frac{6803298487826435051217540}{411340519227716149383203}$ .

A major advancement has been the characterization provided by J.Tunnell. He has proved that if the Birch and Swinnerton-Dyer conjecture is true, then an odd squarefree number $n$ is congruent if and only if the two sets

$\begin{array}{l} \{(a,b,c)\in \mathbf{Z}^3 : 2a^2+b^2+8c^2 = n\,,\mathrm{with\ }c\mathrm{\ odd}\}\\ \{(a,b,c)\in \mathbf{Z}^3 : 2a^2+b^2+8c^2 = n\,,\mathrm{with\ }c\mathrm{\ even} \}\end{array}$

have the same cardinality. A similar relationship holds for even squarefree numbers.

You can download a text file (primitive-congruent-mod123.txt) of 1.6 MB, containing a list of the squarefree congruent numbers below $10^{7}$ . To save further space the numbers of the form $8k+\{5,6,7\}$ which are all congruent, are omitted from the list.

The smallest 3 × 3 magic square whose entries are consecutive congruent numbers is

718	701	711
703	710	717
709	719	702

The first congruent numbers are 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 more terms

Below, the spiral pattern of congruent numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many congruent numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Bill Hart, A Trillion Triangles, American Institute of Mathematics
Wikipedia, Congruent number
OEIS, Sequence A003273

Congruent numbers can also be... (you may click on names or numbers and on + to get more values)

a-pointer 13 101 103 + 9998333 ABA 24 375 384 + 9961472 aban 13 14 15 + 9000999 abundant 20 24 30 + 9999996 Achilles 500 864 1125 + 9984600 admirable 20 24 30 + 9999942 alt.fact. 101 4421 326981 alternating 14 21 23 + 9898989 amenable 13 20 21 + 9999997 amicable 220 284 2620 + 9660950 anti-perfect 244 285 apocalyptic 157 220 222 + 29999 arithmetic 13 14 15 + 9999999 astonishing 15 216 429 + 3882813 automorphic 376 90625 109376 balanced p. 53 157 173 + 9997327 Bell 15 52 877 + 4213597 bemirp 1061 1901 10061 + 1806061 betrothed 1575 1648 1925 + 9345903 binomial 15 20 21 + 9992685 brilliant 14 15 21 + 9999727 c.decagonal 31 61 101 + 9989911 c.heptagonal 22 71 148 + 9978613 c.nonagonal 28 55 136 + 9970345 c.pentagonal 31 141 181 + 9985006 c.square 13 41 61 + 9985981 c.triangular 31 46 85 + 9996214 cake 15 93 299 + 9963072 Canada 125 581 8549 16999 Carmichael 561 1105 2465 + 9585541 Carol 47 223 959 + 4190207 Catalan 14 429 1430 + 9694845 Chen 13 23 29 + 9999991 compositorial 24 17280 constructible 15 20 24 + 8912896 cube 125 216 343 + 9938375 Cullen 65 161 49153 Cunningham 15 24 28 + 9999999 Curzon 14 21 29 + 9999990 cyclic 13 15 23 + 9999997 D-number 15 21 39 + 7043133 d-powerful 24 63 135 + 9999423 de Polignac 127 149 373 + 9999997 decagonal 52 85 126 + 9993501 deceptive 703 2821 2981 + 9863461 deficient 13 14 15 + 9999999 dig.balanced 15 21 37 + 9999996 double fact. 15 384 3840 + 645120 droll 240 9984 14080 + 9461760 Duffinian 21 39 55 + 9999997 eban 30 34 46 + 6066064 economical 13 14 15 + 9999991 emirp 13 31 37 + 9999943 emirpimes 15 39 62 + 9999998 enlightened 2176 2744 119911 + 5117695 equidigital 13 14 15 + 9999991 eRAP 20 24 1104 + 9960047 esthetic 21 23 34 + 9898989 Eulerian 120 247 302 + 2203488 evil 15 20 23 + 9999999 factorial 24 120 720 + 3628800 fibodiv 14 28 47 + 9869390 Fibonacci 13 21 34 + 9227465 Friedman 125 126 127 + 999964 frugal 125 343 1029 + 9998677 gapful 110 120 135 + 9999990 Gilda 29 78 110 + 3606368 Giuga 30 66198 good prime 29 37 41 + 9993317 happy 13 23 28 + 9999992 harmonic 28 270 496 + 8506400 Harshad 20 21 24 + 9999990 heptagonal 34 55 112 + 9987004 hex 37 61 127 + 9997351 hexagonal 15 28 45 + 9988215 highly composite 24 60 120 + 8648640 hoax 22 84 85 + 9999895 Hogben 13 21 31 + 9969807 Honaker 263 457 1039 + 9985231 house 78 271 434 + 9272286 hungry 712201 2401519 7339199 hyperperfect 21 28 301 + 9699181 iban 14 20 21 + 777774 iccanobiF 13 39 124 + 9679838 idoneal 13 15 21 + 1848 inconsummate 62 63 65 + 999978 insolite 111 1122112 interprime 15 21 30 + 9999982 Jacobsthal 21 85 341 + 5592405 Jordan-Polya 24 96 120 + 9953280 junction 101 103 109 + 9999959 Kaprekar 45 55 703 + 9999999 katadrome 20 21 30 + 9876543 Kynea 23 79 287 + 4198399 Lehmer 15 85 133 + 9997351 Leyland 54 145 320 + 9865625 lonely 23 53 120 + 4652430 Lucas 29 47 199 + 4870847 lucky 13 15 21 + 9999997 Lynch-Bell 15 24 124 + 9812376 m-pointer 23 61 1231 + 9111341 magic 15 34 65 + 9951391 magnanimous 14 20 21 + 9959374 metadrome 13 14 15 + 3456789 modest 13 23 29 + 9999999 Moran 21 45 63 + 9999950 Motzkin 21 127 323 + 6536382 narcissistic 371 407 54748 + 1741725 nialpdrome 20 21 22 + 9999999 nonagonal 24 46 111 + 9992125 nude 15 22 24 + 9999999 O'Halloran 20 60 84 + 924 oban 13 15 20 + 999 octagonal 21 65 96 + 9977280 odious 13 14 21 + 9999998 Ormiston 18397 19013 19031 + 9994631 palindromic 22 55 77 + 9999999 palprime 101 151 181 + 9978799 pancake 22 29 37 + 9997157 panconsummate 14 15 20 + 1093 pandigital 15 21 78 + 9998037 partition 15 22 30 + 8118264 pentagonal 22 70 92 + 9983310 perfect 28 496 8128 pernicious 13 14 20 + 9999998 Perrin 22 29 39 + 9141872 Pierpont 13 37 109 + 5308417 plaindrome 13 14 15 + 9999999 Poulet 341 561 645 + 9995671 power 125 216 343 + 9938375 powerful 125 216 343 + 9984600 practical 20 24 28 + 9999990 prim.abundant 20 30 56 + 9999942 prime 13 23 29 + 9999991 primeval 13 37 137 + 1234679 primorial 30 210 2310 + 9699690 pronic 20 30 56 + 9988760 Proth 13 41 65 + 9879553 pseudoperfect 20 24 28 + 999999 rare 65 repdigit 22 55 77 + 9999999 repfigit 14 28 47 + 7913837 repunit 13 15 21 + 9969807 Rhonda 5265 5439 8526 + 9843255 Ruth-Aaron 15 24 77 + 9981936 Saint-Exupery 60 480 1620 + 9903180 Sastry 183 40495 self 20 31 53 + 9999998 self-describing 22 4444 224444 + 666666 semiprime 14 15 21 + 9999998 sliding 20 29 52 + 9250000 Smith 22 85 94 + 9999895 Sophie Germain 23 29 41 + 9999653 sphenic 30 70 78 + 9999983 star 13 37 181 + 9992341 straight-line 111 135 159 + 9999999 strobogrammatic 69 88 96 + 9968966 strong prime 29 37 41 + 9999901 subfactorial 265 1854 133496 super Niven 20 24 30 + 9990000 super-d 31 69 119 + 9999981 superabundant 24 60 120 + 8648640 tau 24 56 60 + 9999992 taxicab 13832 110808 134379 + 9560896 tcefrep 498906 tetrahedral 20 56 84 + 9962680 tetranacci 15 29 56 + 7555935 triangular 15 21 28 + 9992685 tribonacci 13 24 149 + 8646064 trimorphic 24 125 375 + 9999999 truncatable prime 13 23 29 + 9986113 twin 13 29 31 + 9999973 uban 13 15 20 + 9000095 Ulam 13 28 38 + 9999999 undulating 101 141 151 + 9898989 unprimeable 206 208 320 + 9999988 untouchable 52 88 96 + 999996 upside-down 28 37 46 + 9995111 vampire 6880 102510 105264 + 939658 wasteful 20 22 24 + 9999999 weak prime 13 23 31 + 9999991 weakly prime 505447 584141 971767 + 9931447 weird 70 4030 5830 + 999670 Wieferich 1093 3279 3511 + 6161805 Woodall 23 63 80 + 9961471 Zeisel 1419 1885 5719 + 9271805 Zuckerman 15 24 111 + 9813312 Zumkeller 20 24 28 + 99996 zygodrome 22 55 77 + 9999999