A number which is the area of a right triangle with rational sides. more
The first 600 congruent numbers :
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,
124,
125,
126,
127,
133,
134,
135,
136,
137,
138,
141,
142,
143,
145,
148,
149,
150,
151,
152,
154,
156,
157,
158,
159,
161,
164,
165,
166,
167,
173,
174,
175,
180,
181,
182,
183,
184,
188,
189,
190,
191,
194,
197,
198,
199,
205,
206,
207,
208,
210,
212,
213,
214,
215,
216,
219,
220,
221,
222,
223,
224,
226,
229,
230,
231,
237,
238,
239,
240,
244,
245,
246,
247,
248,
252,
253,
254,
255,
257,
260,
261,
262,
263,
265,
269,
270,
271,
276,
277,
278,
279,
280,
284,
285,
286,
287,
291,
293,
294,
295,
299,
301,
302,
303,
306,
308,
309,
310,
311,
312,
313,
316,
317,
318,
319,
320,
323,
325,
326,
327,
330,
333,
334,
335,
336,
340,
341,
342,
343,
344,
348,
349,
350,
351,
352,
353,
357,
358,
359,
365,
366,
367,
368,
369,
371,
372,
373,
374,
375,
376,
380,
381,
382,
383,
384,
386,
389,
390,
391,
395,
397,
398,
399,
404,
405,
406,
407,
408,
410,
412,
413,
414,
415,
421,
422,
423,
426,
429,
430,
431,
434,
436,
437,
438,
439,
440,
442,
444,
445,
446,
447,
448,
453,
454,
455,
457,
461,
462,
463,
464,
465,
468,
469,
470,
471,
472,
476,
477,
478,
479,
480,
485,
486,
487,
493,
494,
495,
496,
500,
501,
502,
503,
504,
505,
508,
509,
510,
511,
514,
517,
518,
519,
525,
526,
527,
532,
533,
534,
535,
536,
540,
541,
542,
543,
544,
546,
548,
549,
550,
551,
552,
557,
558,
559,
561,
564,
565,
566,
567,
568,
572,
573,
574,
575,
580,
581,
582,
583,
585,
589,
590,
591,
592,
596,
597,
598,
599,
600,
602,
604,
605,
606,
607,
608,
609,
613,
614,
615,
616,
621,
622,
623,
624,
628,
629,
630,
631,
632,
636,
637,
638,
639,
644,
645,
646,
647,
651,
653,
654,
655,
656,
658,
660,
661,
662,
663,
664,
668,
669,
670,
671,
674,
677,
678,
679,
685,
686,
687,
689,
692,
693,
694,
695,
696,
700,
701,
702,
703,
709,
710,
711,
717,
718,
719,
720,
721,
723,
724,
725,
726,
727,
728,
731,
732,
733,
734,
735,
736,
741,
742,
743,
749,
750,
751,
752,
756,
757,
758,
759,
760,
761,
764,
765,
766,
767,
773,
774,
775,
776,
777,
781,
782,
783,
788,
789,
790,
791,
792,
793,
796,
797,
798,
799,
805,
806,
807,
813,
814,
815,
820,
821,
822,
823,
824,
828,
829,
830,
831,
832,
837,
838,
839,
840,
845,
846,
847,
848,
850,
852,
853,
854,
855,
856,
860,
861,
862,
863,
864,
866,
869,
870,
871,
876,
877,
878,
879,
880,
884,
885,
886,
887,
888,
889,
890,
892,
893,
894,
895,
896,
901,
902,
903,
904,
905,
909,
910,
911,
915,
916,
917,
918,
919,
920,
924,
925,
926,
927,
933,
934,
935,
941,
942,
943,
948,
949,
950,
951,
952,
956,
957,
958,
959,
960,
965,
966,
967,
973,
974,
975,
976,
980,
981,
982,
983,
984,
985,
987,
988,
989,
990,
991,
992,
995,
997,
998,
999,
1003,
1005,
1006,
1007,
1008,
1012,
1013,
1014,
1015,
1016,
1020,
1021,
1022,
1023,
1025,
1028,
1029,
1030,
1031,
1037,
1038,
1039,
1040,
1044,
1045,
1046,
1047,
1048,
1052,
1053,
1054,
1055.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 5410361 values, from 5 to 9999999).
n\r | 0 | 1 |
2 | 2725293 | 2685068 | 2 |
3 | 1829985 | 1790572 | 1789804 | 3 |
4 | 1370461 | 1357518 | 1354832 | 1327550 | 4 |
5 | 1117964 | 1085801 | 1060544 | 1060174 | 1085878 | 5 |
6 | 923391 | 889487 | 900817 | 906594 | 901085 | 888987 | 6 |
7 | 807086 | 767159 | 767116 | 767133 | 766934 | 767536 | 767397 | 7 |
8 | 690746 | 107518 | 104832 | 77550 | 679715 | 1250000 | 1250000 | 1250000 | 8 |
9 | 613997 | 596972 | 596732 | 610137 | 596776 | 596421 | 605851 | 596824 | 596651 | 9 |
10 | 563765 | 538581 | 533363 | 526893 | 547664 | 554199 | 547220 | 527181 | 533281 | 538214 | 10 |
11 | 504913 | 490604 | 490336 | 490647 | 490416 | 490331 | 490580 | 490529 | 490692 | 490625 | 490688 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.