Following are the 104 primes less than 1000 for which a concordance
solution is known. Most of these results (especially those with
very large numbers) were computed by David Einstein, including the
case of p=863. Allan MacLeod found the same solution for p=863,
and also found a solution for p=983 in Sept 97, thereby completing
the table.
The table lists two sets of values of r,s for each prime. Using
the left hand set the values of a,b,c,d such that
b^2 + d^2 = c^2 b^2 + pd^2 = a^2
can be computed using the formulas
v=r^2 u = s^2 - r^2
and
ka = u^2 + 2uv - (p-1)v^2
kb = u^2 - 2(p-1)uv - (p-1)v^2
kc = u^2 + (p-1)v^2
kd = sqrt[ 4uv(u+v)(pv-(u+v)) ]
Using the righthand values of r,s the values of a,b,c,d can
be computed using the formulas
v=r^2 u = s^2 + r^2
and
ka = u^2 - 2uv - (p-1)v^2
kb = u^2 + 2(p-1)uv - (p-1)v^2
kc = u^2 + (p-1)v^2
kd = sqrt[ 4uv(u-v)(pv+(u-v)) ]
Note that the gcd represented by k in these equations must be a
divisor of p-1.
(s^2 - r^2)(pr^2 - s^2) (s^2 + r^2)(pr^2 + s^2)
is a square (uv > 0) is a square (uv < 0)
----------------------- -----------------------
p r s r s
---- ------- -------- ------- -------
7 1 2 1 1
11 1 3 2 1
17 1 3 1 1
23 5 6 1 7
31 1 2 1 1
41 1 3 1 2
47 13 36 7 17
53 5 13 6 17
59 1 3 2 5
61 1 7 2 1
71 1 6 1 1
79 1 2 1 5
83 17 33 20 107
97 1 7 1 1
101 1 9 2 1
107 1 3 2 7
113 5 9 1 7
127 1 8 1 1
137 1 3 1 4
149 13 85 42 67
151 1 2 2 1
157 1 7 2 3
167 17 66 7 23
179 89 153 88 835
181 5 13 2 11
193 5 11 1 7
199 1 10 1 1
211 1 13 2 1
227 41 297 208 383
233 17 33 10 91
239 1 8 4 1
241 1 5 1 1
251 1 3 2 11
257 17 81 7 23
263 545 5874
281 1 15 2 1
293 2465 10657
307 1 17 4 1
313 5 7 1 18
331 5 17 2 11
337 1 13 1 1
347 1 3 2 13
349 1 7 2 5
353 5 93 7 1
359 16481 35250
367 1 2 1 11
383 38785 222336
389 1073 1105
401 1 3 1 2
409 1 5
421 5 19
433 5 103
449 1 15
461 29 213
463 5 16
467 30161 64017
479 25 264
487 97 146
491 193 291
503 845 11454
521 1 3
523 257 1507
541 13 229
547 1 13
563 113 1137
571 13 25
577 1 17
587 1 3
599 1 24
601 1 11
613 25 313
631 1 8
647 5 14
661 1 23
673 5 19
677 29 133
691 17 259
701 13 27
719 4663525 81621996
727 13 134
733 16237 94213
739 1 17
751 1 4
761 17 21
769 29 35
773 144625 677137
809 1 3
811 41 283
823 1 26
839 64025 941994
859 5 49
863 2365498105 13810017384 655373999 3280495007 (einstein/macleod)
877 13 19
881 1 21
887 68140133 239252694
911 1 6
919 1 26
937 1 17
941 653 1029
953 25 39
967 1 22
977 1 3
983 2917382885 59634234294 3129972023 2688032911 (macleod)
991 1 10
These are the 48 primes less than 1000 that are definitely
discordant, based on my elementary proof (and in agreement
with the Birch/Swinnerton-Dyer conjecture and numerical
results):
2 3 5 13 19 29 37 43
67 73 89 109 139 163 173 197
229 269 277 283 317 373 379 397
419 457 499 509 557 569 617 619
643 653 659 683 709 757 787 797
827 829 853 857 883 907 947 997
These are the 16 primes less than 1000 that are not ruled out
by my proof, but are "ruled out" on the assumption of the
Birch/Swinnerton-Dyer conjecture:
103 131 191 223 271 311 431 439
443 593 607 641 743 821 929 971