```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

```