## p^2 + 1 = q^2 + r^2 with (p,q,r) prime numbers

$p^2 \; + \; 1 \; = \; q^2 \; + \; r^2$

with primes   $p, \; q$,   and   $r$

Solutions for all the prime numbers less than 1000 are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

$13^2 \; + \; 1 \; = \; 7^2 \; + \; 11^2$

$23^2 \; + 1 \; = 13^2 \; + \; 19^2$
$43^2 \; + 1 \; = 13^2 \; + \; 41^2$
$53^2 \; + 1 \; = 31^2 \; + \; 43^2$
$73^2 \; + 1 \; = 43^2 \; + \; 59^2$

$103^2 \; + \; 1 \; = \; 61^2 \; + \; 83^2$
$163^2 \; + \; 1 \; = \; 97^2 \; + \; 131^2$

$173^2 \; + \; 1 \; = \; 103^2 \; + \; 139^2$
$263^2 \; + \; 1 \; = \; 157^2 \; + \; 211^2$
$353^2 \; + \; 1 \; = \; 211^2 \; + \; 283^2$
$383^2 \; + \; 1 \; = \; 229^2 \; + \; 307^2$
$523^2 \; + \; 1 \; = \; 313^2 \; + \; 419^2$
$613^2 \; + \; 1 \; = \; 367^2 \; + \; 491^2$
$683^2 \; + \; 1 \; = \; 409^2 \; + \; 547^2$
$733^2 \; + \; 1 \; = \; 439^2 \; + \; 587^2$
$773^2 \; + \; 1 \; = \; 463^2 \; + \; 619^2$