"If a number is the sum of two consecutive square numbers, and it is not sum of any other two square numbers, then the number is prime. And in reverse, if the sum of two consecutive square numbers is prime, then it is not the sum of two other square numbers (where 0 is considered a square number)."
After I republished those earlier blog posts in this newer blog, that statement began to interest me again, so I took the time to think on it. I checked the conjecture with a C computer program for small numbers (i.e. until 50000²+50001²), and I also generalized the conjecture (and also checked that for some numbers) in the following way:
"Let S be the sum of an even and an odd square number relatively prime to each other. S is prime if and only if S is not sum of any other two square numbers (other than used in its definition)."
Here is some output of a C program I checked this conjecture with:
2²+ 1²= 5 (prime) (ok)
4²+ 1²= 17 (prime) (ok)
4²+ 3²= 25= 5* 5= 0²+ 5²
6²+ 1²= 37 (prime) (ok)
6²+ 5²= 61 (prime) (ok)
8²+ 1²= 65= 5* 13= 4²+ 7²
8²+ 3²= 73 (prime) (ok)
8²+ 5²= 89 (prime) (ok)
8²+ 7²= 113 (prime) (ok)
10²+ 1²= 101 (prime) (ok)
10²+ 3²= 109 (prime) (ok)
10²+ 7²= 149 (prime) (ok)
10²+ 9²= 181 (prime) (ok)
12²+ 1²= 145= 5* 29= 8²+ 9²
12²+ 5²= 169= 13* 13= 0²+ 13²
12²+ 7²= 193 (prime) (ok)
12²+ 11²= 265= 5* 53= 3²+ 16²
14²+ 1²= 197 (prime) (ok)
14²+ 3²= 205= 5* 41= 6²+ 13²
14²+ 5²= 221= 13* 17= 10²+ 11²
14²+ 9²= 277 (prime) (ok)
14²+ 11²= 317 (prime) (ok)
14²+ 13²= 365= 5* 73= 2²+ 19²
16²+ 1²= 257 (prime) (ok)
16²+ 3²= 265= 5* 53= 11²+ 12²
16²+ 5²= 281 (prime) (ok)
16²+ 7²= 305= 5* 61= 4²+ 17²
16²+ 9²= 337 (prime) (ok)
16²+ 11²= 377= 13* 29= 4²+ 19²
16²+ 13²= 425= 5* 85= 5²+ 20²
16²+ 15²= 481= 13* 37= 9²+ 20²
18²+ 1²= 325= 5* 65= 6²+ 17²
18²+ 5²= 349 (prime) (ok)
18²+ 7²= 373 (prime) (ok)
18²+ 11²= 445= 5* 89= 2²+ 21²
18²+ 13²= 493= 17* 29= 3²+ 22²
18²+ 17²= 613 (prime) (ok)
20²+ 1²= 401 (prime) (ok)
20²+ 3²= 409 (prime) (ok)
20²+ 7²= 449 (prime) (ok)
20²+ 9²= 481= 13* 37= 15²+ 16²
20²+ 11²= 521 (prime) (ok)
20²+ 13²= 569 (prime) (ok)
20²+ 17²= 689= 13* 53= 8²+ 25²
20²+ 19²= 761 (prime) (ok)
22²+ 1²= 485= 5* 97= 14²+ 17²
22²+ 3²= 493= 17* 29= 13²+ 18²
22²+ 5²= 509 (prime) (ok)
22²+ 7²= 533= 13* 41= 2²+ 23²
22²+ 9²= 565= 5* 113= 6²+ 23²
22²+ 13²= 653 (prime) (ok)
22²+ 15²= 709 (prime) (ok)
22²+ 17²= 773 (prime) (ok)
22²+ 19²= 845= 5* 169= 2²+ 29²
22²+ 21²= 925= 5* 185= 5²+ 30²
24²+ 1²= 577 (prime) (ok)
24²+ 5²= 601 (prime) (ok)
24²+ 7²= 625= 5* 125= 0²+ 25²
24²+ 11²= 697= 17* 41= 16²+ 21²
24²+ 13²= 745= 5* 149= 4²+ 27²
24²+ 17²= 865= 5* 173= 9²+ 28²
24²+ 19²= 937 (prime) (ok)
24²+ 23²= 1105= 5* 221= 4²+ 33²
26²+ 1²= 677 (prime) (ok)
26²+ 3²= 685= 5* 137= 18²+ 19²
26²+ 5²= 701 (prime) (ok)
26²+ 7²= 725= 5* 145= 10²+ 25²
26²+ 9²= 757 (prime) (ok)
26²+ 11²= 797 (prime) (ok)
26²+ 15²= 901= 17* 53= 1²+ 30²
26²+ 17²= 965= 5* 193= 2²+ 31²
26²+ 19²= 1037= 17* 61= 14²+ 29²
26²+ 21²= 1117 (prime) (ok)
26²+ 23²= 1205= 5* 241= 7²+ 34²
26²+ 25²= 1301 (prime) (ok)
28²+ 1²= 785= 5* 157= 16²+ 23²
28²+ 3²= 793= 13* 61= 8²+ 27²
28²+ 5²= 809 (prime) (ok)
28²+ 9²= 865= 5* 173= 17²+ 24²
28²+ 11²= 905= 5* 181= 8²+ 29²
28²+ 13²= 953 (prime) (ok)
28²+ 15²= 1009 (prime) (ok)
28²+ 17²= 1073= 29* 37= 7²+ 32²
28²+ 19²= 1145= 5* 229= 11²+ 32²
28²+ 23²= 1313= 13* 101= 17²+ 32²
28²+ 25²= 1409 (prime) (ok)
28²+ 27²= 1513= 17* 89= 12²+ 37²
30²+ 1²= 901= 17* 53= 15²+ 26²
30²+ 7²= 949= 13* 73= 18²+ 25²
30²+ 11²= 1021 (prime) (ok)
30²+ 13²= 1069 (prime) (ok)
30²+ 17²= 1189= 29* 41= 10²+ 33²
30²+ 19²= 1261= 13* 97= 6²+ 35²
30²+ 23²= 1429 (prime) (ok)
30²+ 29²= 1741 (prime) (ok)
32²+ 1²= 1025= 5* 205= 8²+ 31²
32²+ 3²= 1033 (prime) (ok)
32²+ 5²= 1049 (prime) (ok)
32²+ 7²= 1073= 29* 37= 17²+ 28²
32²+ 9²= 1105= 5* 221= 4²+ 33²
32²+ 11²= 1145= 5* 229= 19²+ 28²
32²+ 13²= 1193 (prime) (ok)
32²+ 15²= 1249 (prime) (ok)
32²+ 17²= 1313= 13* 101= 23²+ 28²
32²+ 19²= 1385= 5* 277= 4²+ 37²
32²+ 21²= 1465= 5* 293= 13²+ 36²
32²+ 23²= 1553 (prime) (ok)
32²+ 25²= 1649= 17* 97= 7²+ 40²
32²+ 27²= 1753 (prime) (ok)
32²+ 29²= 1865= 5* 373= 4²+ 43²
32²+ 31²= 1985= 5* 397= 7²+ 44²
34²+ 1²= 1157= 13* 89= 14²+ 31²
34²+ 3²= 1165= 5* 233= 18²+ 29²
34²+ 5²= 1181 (prime) (ok)
34²+ 7²= 1205= 5* 241= 23²+ 26²
34²+ 9²= 1237 (prime) (ok)
34²+ 11²= 1277 (prime) (ok)
34²+ 13²= 1325= 5* 265= 10²+ 35²
34²+ 15²= 1381 (prime) (ok)
34²+ 19²= 1517= 37* 41= 26²+ 29²
34²+ 21²= 1597 (prime) (ok)
34²+ 23²= 1685= 5* 337= 2²+ 41²
34²+ 25²= 1781= 13* 137= 10²+ 41²
34²+ 27²= 1885= 5* 377= 6²+ 43²
34²+ 29²= 1997 (prime) (ok)
34²+ 31²= 2117= 29* 73= 1²+ 46²
34²+ 33²= 2245= 5* 449= 6²+ 47²
36²+ 1²= 1297 (prime) (ok)
36²+ 5²= 1321 (prime) (ok)
36²+ 7²= 1345= 5* 269= 16²+ 33²
36²+ 11²= 1417= 13* 109= 24²+ 29²
36²+ 13²= 1465= 5* 293= 21²+ 32²
36²+ 17²= 1585= 5* 317= 8²+ 39²
36²+ 19²= 1657 (prime) (ok)
36²+ 23²= 1825= 5* 365= 12²+ 41²
36²+ 25²= 1921= 17* 113= 20²+ 39²
36²+ 29²= 2137 (prime) (ok)
36²+ 31²= 2257= 37* 61= 24²+ 41²
36²+ 35²= 2521 (prime) (ok)
38²+ 1²= 1445= 5* 289= 17²+ 34²
38²+ 3²= 1453 (prime) (ok)
38²+ 5²= 1469= 13* 113= 10²+ 37²
38²+ 7²= 1493 (prime) (ok)
38²+ 9²= 1525= 5* 305= 2²+ 39²
38²+ 11²= 1565= 5* 313= 14²+ 37²
38²+ 13²= 1613 (prime) (ok)
38²+ 15²= 1669 (prime) (ok)
38²+ 17²= 1733 (prime) (ok)
38²+ 21²= 1885= 5* 377= 6²+ 43²
38²+ 23²= 1973 (prime) (ok)
38²+ 25²= 2069 (prime) (ok)
38²+ 27²= 2173= 41* 53= 18²+ 43²
38²+ 29²= 2285= 5* 457= 13²+ 46²
38²+ 31²= 2405= 5* 481= 2²+ 49²
38²+ 33²= 2533= 17* 149= 18²+ 47²
38²+ 35²= 2669= 17* 157= 13²+ 50²
38²+ 37²= 2813= 29* 97= 2²+ 53²
40²+ 1²= 1601 (prime) (ok)
40²+ 3²= 1609 (prime) (ok)
40²+ 7²= 1649= 17* 97= 25²+ 32²
40²+ 9²= 1681= 41* 41= 0²+ 41²
40²+ 11²= 1721 (prime) (ok)
40²+ 13²= 1769= 29* 61= 20²+ 37²
40²+ 17²= 1889 (prime) (ok)
40²+ 19²= 1961= 37* 53= 5²+ 44²
40²+ 21²= 2041= 13* 157= 4²+ 45²
40²+ 23²= 2129 (prime) (ok)
40²+ 27²= 2329= 17* 137= 5²+ 48²
40²+ 29²= 2441 (prime) (ok)
40²+ 31²= 2561= 13* 197= 25²+ 44²
40²+ 33²= 2689 (prime) (ok)
40²+ 37²= 2969 (prime) (ok)
40²+ 39²= 3121 (prime) (ok)
42²+ 1²= 1765= 5* 353= 26²+ 33²
42²+ 5²= 1789 (prime) (ok)
42²+ 11²= 1885= 5* 377= 6²+ 43²
42²+ 13²= 1933 (prime) (ok)
42²+ 17²= 2053 (prime) (ok)
42²+ 19²= 2125= 5* 425= 3²+ 46²
42²+ 23²= 2293 (prime) (ok)
42²+ 25²= 2389 (prime) (ok)
42²+ 29²= 2605= 5* 521= 2²+ 51²
42²+ 31²= 2725= 5* 545= 15²+ 50²
42²+ 37²= 3133= 13* 241= 18²+ 53²
42²+ 41²= 3445= 5* 689= 9²+ 58²
44²+ 1²= 1937= 13* 149= 16²+ 41²
44²+ 3²= 1945= 5* 389= 24²+ 37²
44²+ 5²= 1961= 37* 53= 19²+ 40²
44²+ 7²= 1985= 5* 397= 31²+ 32²
44²+ 9²= 2017 (prime) (ok)
44²+ 13²= 2105= 5* 421= 16²+ 43²
44²+ 15²= 2161 (prime) (ok)
44²+ 17²= 2225= 5* 445= 4²+ 47²
44²+ 19²= 2297 (prime) (ok)
44²+ 21²= 2377 (prime) (ok)
44²+ 23²= 2465= 5* 493= 8²+ 49²
44²+ 25²= 2561= 13* 197= 31²+ 40²
44²+ 27²= 2665= 5* 533= 8²+ 51²
44²+ 29²= 2777 (prime) (ok)
44²+ 31²= 2897 (prime) (ok)
44²+ 35²= 3161= 29* 109= 5²+ 56²
44²+ 37²= 3305= 5* 661= 13²+ 56²
44²+ 39²= 3457 (prime) (ok)
44²+ 41²= 3617 (prime) (ok)
44²+ 43²= 3785= 5* 757= 8²+ 61²
46²+ 1²= 2117= 29* 73= 31²+ 34²
46²+ 3²= 2125= 5* 425= 10²+ 45²
46²+ 5²= 2141 (prime) (ok)
46²+ 7²= 2165= 5* 433= 22²+ 41²
46²+ 9²= 2197= 13* 169= 26²+ 39²
46²+ 11²= 2237 (prime) (ok)
46²+ 13²= 2285= 5* 457= 29²+ 38²
46²+ 15²= 2341 (prime) (ok)
46²+ 17²= 2405= 5* 481= 2²+ 49²
46²+ 19²= 2477 (prime) (ok)
46²+ 21²= 2557 (prime) (ok)
46²+ 25²= 2741 (prime) (ok)
46²+ 27²= 2845= 5* 569= 6²+ 53²
46²+ 29²= 2957 (prime) (ok)
46²+ 31²= 3077= 17* 181= 26²+ 49²
46²+ 33²= 3205= 5* 641= 17²+ 54²
46²+ 35²= 3341= 13* 257= 29²+ 50²
46²+ 37²= 3485= 5* 697= 2²+ 59²
46²+ 39²= 3637 (prime) (ok)
46²+ 41²= 3797 (prime) (ok)
46²+ 43²= 3965= 5* 793= 11²+ 62²
46²+ 45²= 4141= 41* 101= 35²+ 54²
48²+ 1²= 2305= 5* 461= 28²+ 39²
48²+ 5²= 2329= 17* 137= 27²+ 40²
48²+ 7²= 2353= 13* 181= 12²+ 47²
48²+ 11²= 2425= 5* 485= 20²+ 45²
48²+ 13²= 2473 (prime) (ok)
48²+ 17²= 2593 (prime) (ok)
48²+ 19²= 2665= 5* 533= 8²+ 51²
48²+ 23²= 2833 (prime) (ok)
48²+ 25²= 2929= 29* 101= 15²+ 52²
48²+ 29²= 3145= 5* 629= 3²+ 56²
48²+ 31²= 3265= 5* 653= 4²+ 57²
48²+ 35²= 3529 (prime) (ok)
48²+ 37²= 3673 (prime) (ok)
48²+ 41²= 3985= 5* 797= 4²+ 63²
48²+ 43²= 4153 (prime) (ok)
48²+ 47²= 4513 (prime) (ok)
50²+ 1²= 2501= 41* 61= 10²+ 49²
50²+ 3²= 2509= 13* 193= 22²+ 45²
50²+ 7²= 2549 (prime) (ok)
50²+ 9²= 2581= 29* 89= 30²+ 41²
50²+ 11²= 2621 (prime) (ok)
50²+ 13²= 2669= 17* 157= 35²+ 38²
50²+ 17²= 2789 (prime) (ok)
50²+ 19²= 2861 (prime) (ok)
50²+ 21²= 2941= 17* 173= 5²+ 54²
50²+ 23²= 3029= 13* 233= 2²+ 55²
50²+ 27²= 3229 (prime) (ok)
50²+ 29²= 3341= 13* 257= 35²+ 46²
50²+ 31²= 3461 (prime) (ok)
50²+ 33²= 3589= 37* 97= 15²+ 58²
50²+ 37²= 3869= 53* 73= 5²+ 62²
50²+ 39²= 4021 (prime) (ok)
50²+ 41²= 4181= 37* 113= 34²+ 55²
50²+ 43²= 4349 (prime) (ok)
50²+ 47²= 4709= 17* 277= 22²+ 65²
50²+ 49²= 4901= 13* 377= 1²+ 70²
(Edited.) NOTE: there can be other tests for the case when the odd number is greater than the even one:
3²+ 2²= 13 (prime) (ok)
5²+ 2²= 29 (prime) (ok)
5²+ 4²= 41 (prime) (ok)
7²+ 2²= 53 (prime) (ok)
7²+ 4²= 65= 5* 13= 1²+ 8²
7²+ 6²= 85= 5* 17= 2²+ 9²
9²+ 2²= 85= 5* 17= 6²+ 7²
9²+ 4²= 97 (prime) (ok)
9²+ 8²= 145= 5* 29= 1²+ 12²
11²+ 2²= 125= 5* 25= 5²+ 10²
11²+ 4²= 137 (prime) (ok)
11²+ 6²= 157 (prime) (ok)
11²+ 8²= 185= 5* 37= 4²+ 13²
11²+ 10²= 221= 13* 17= 5²+ 14²
13²+ 2²= 173 (prime) (ok)
13²+ 4²= 185= 5* 37= 8²+ 11²
13²+ 6²= 205= 5* 41= 3²+ 14²
13²+ 8²= 233 (prime) (ok)
13²+ 10²= 269 (prime) (ok)
13²+ 12²= 313 (prime) (ok)
15²+ 2²= 229 (prime) (ok)
15²+ 4²= 241 (prime) (ok)
15²+ 8²= 289= 17* 17= 0²+ 17²
15²+ 14²= 421 (prime) (ok)
17²+ 2²= 293 (prime) (ok)
17²+ 4²= 305= 5* 61= 7²+ 16²
17²+ 6²= 325= 5* 65= 1²+ 18²
17²+ 8²= 353 (prime) (ok)
17²+ 10²= 389 (prime) (ok)
17²+ 12²= 433 (prime) (ok)
17²+ 14²= 485= 5* 97= 1²+ 22²
17²+ 16²= 545= 5* 109= 4²+ 23²
19²+ 2²= 365= 5* 73= 13²+ 14²
19²+ 4²= 377= 13* 29= 11²+ 16²
19²+ 6²= 397 (prime) (ok)
19²+ 8²= 425= 5* 85= 5²+ 20²
19²+ 10²= 461 (prime) (ok)
19²+ 12²= 505= 5* 101= 8²+ 21²
19²+ 14²= 557 (prime) (ok)
19²+ 16²= 617 (prime) (ok)
19²+ 18²= 685= 5* 137= 3²+ 26²
21²+ 2²= 445= 5* 89= 11²+ 18²
21²+ 4²= 457 (prime) (ok)
21²+ 8²= 505= 5* 101= 12²+ 19²
21²+ 10²= 541 (prime) (ok)
21²+ 16²= 697= 17* 41= 11²+ 24²
21²+ 20²= 841= 29* 29= 0²+ 29²
23²+ 2²= 533= 13* 41= 7²+ 22²
23²+ 4²= 545= 5* 109= 16²+ 17²
23²+ 6²= 565= 5* 113= 9²+ 22²
23²+ 8²= 593 (prime) (ok)
23²+ 10²= 629= 17* 37= 2²+ 25²
23²+ 12²= 673 (prime) (ok)
23²+ 14²= 725= 5* 145= 7²+ 26²
23²+ 16²= 785= 5* 157= 1²+ 28²
23²+ 18²= 853 (prime) (ok)
23²+ 20²= 929 (prime) (ok)
23²+ 22²= 1013 (prime) (ok)
25²+ 2²= 629= 17* 37= 10²+ 23²
25²+ 4²= 641 (prime) (ok)
25²+ 6²= 661 (prime) (ok)
25²+ 8²= 689= 13* 53= 17²+ 20²
25²+ 12²= 769 (prime) (ok)
25²+ 14²= 821 (prime) (ok)
25²+ 16²= 881 (prime) (ok)
25²+ 18²= 949= 13* 73= 7²+ 30²
25²+ 22²= 1109 (prime) (ok)
25²+ 24²= 1201 (prime) (ok)
27²+ 2²= 733 (prime) (ok)
27²+ 4²= 745= 5* 149= 13²+ 24²
27²+ 8²= 793= 13* 61= 3²+ 28²
27²+ 10²= 829 (prime) (ok)
27²+ 14²= 925= 5* 185= 5²+ 30²
27²+ 16²= 985= 5* 197= 12²+ 29²
27²+ 20²= 1129 (prime) (ok)
27²+ 22²= 1213 (prime) (ok)
27²+ 26²= 1405= 5* 281= 6²+ 37²
29²+ 2²= 845= 5* 169= 13²+ 26²
29²+ 4²= 857 (prime) (ok)
29²+ 6²= 877 (prime) (ok)
29²+ 8²= 905= 5* 181= 11²+ 28²
29²+ 10²= 941 (prime) (ok)
29²+ 12²= 985= 5* 197= 16²+ 27²
29²+ 14²= 1037= 17* 61= 19²+ 26²
29²+ 16²= 1097 (prime) (ok)
29²+ 18²= 1165= 5* 233= 3²+ 34²
29²+ 20²= 1241= 17* 73= 4²+ 35²
29²+ 22²= 1325= 5* 265= 10²+ 35²
29²+ 24²= 1417= 13* 109= 11²+ 36²
29²+ 26²= 1517= 37* 41= 19²+ 34²
29²+ 28²= 1625= 5* 325= 5²+ 40²
31²+ 2²= 965= 5* 193= 17²+ 26²
31²+ 4²= 977 (prime) (ok)
31²+ 6²= 997 (prime) (ok)
31²+ 8²= 1025= 5* 205= 1²+ 32²
31²+ 10²= 1061 (prime) (ok)
31²+ 12²= 1105= 5* 221= 4²+ 33²
31²+ 14²= 1157= 13* 89= 1²+ 34²
31²+ 16²= 1217 (prime) (ok)
31²+ 18²= 1285= 5* 257= 14²+ 33²
31²+ 20²= 1361 (prime) (ok)
31²+ 22²= 1445= 5* 289= 1²+ 38²
31²+ 24²= 1537= 29* 53= 4²+ 39²
31²+ 26²= 1637 (prime) (ok)
31²+ 28²= 1745= 5* 349= 8²+ 41²
31²+ 30²= 1861 (prime) (ok)
33²+ 2²= 1093 (prime) (ok)
33²+ 4²= 1105= 5* 221= 9²+ 32²
33²+ 8²= 1153 (prime) (ok)
33²+ 10²= 1189= 29* 41= 17²+ 30²
33²+ 14²= 1285= 5* 257= 18²+ 31²
33²+ 16²= 1345= 5* 269= 7²+ 36²
33²+ 20²= 1489 (prime) (ok)
33²+ 26²= 1765= 5* 353= 1²+ 42²
33²+ 28²= 1873 (prime) (ok)
33²+ 32²= 2113 (prime) (ok)
35²+ 2²= 1229 (prime) (ok)
35²+ 4²= 1241= 17* 73= 20²+ 29²
35²+ 6²= 1261= 13* 97= 19²+ 30²
35²+ 8²= 1289 (prime) (ok)
35²+ 12²= 1369= 37* 37= 0²+ 37²
35²+ 16²= 1481 (prime) (ok)
35²+ 18²= 1549 (prime) (ok)
35²+ 22²= 1709 (prime) (ok)
35²+ 24²= 1801 (prime) (ok)
35²+ 26²= 1901 (prime) (ok)
35²+ 32²= 2249= 13* 173= 20²+ 43²
35²+ 34²= 2381 (prime) (ok)
37²+ 2²= 1373 (prime) (ok)
37²+ 4²= 1385= 5* 277= 19²+ 32²
37²+ 6²= 1405= 5* 281= 26²+ 27²
37²+ 8²= 1433 (prime) (ok)
37²+ 10²= 1469= 13* 113= 5²+ 38²
37²+ 12²= 1513= 17* 89= 27²+ 28²
37²+ 14²= 1565= 5* 313= 11²+ 38²
37²+ 16²= 1625= 5* 325= 5²+ 40²
37²+ 18²= 1693 (prime) (ok)
37²+ 20²= 1769= 29* 61= 13²+ 40²
37²+ 22²= 1853= 17* 109= 2²+ 43²
37²+ 24²= 1945= 5* 389= 3²+ 44²
37²+ 26²= 2045= 5* 409= 14²+ 43²
37²+ 28²= 2153 (prime) (ok)
37²+ 30²= 2269 (prime) (ok)
37²+ 32²= 2393 (prime) (ok)
37²+ 34²= 2525= 5* 505= 5²+ 50²
37²+ 36²= 2665= 5* 533= 8²+ 51²
39²+ 2²= 1525= 5* 305= 9²+ 38²
39²+ 4²= 1537= 29* 53= 24²+ 31²
39²+ 8²= 1585= 5* 317= 17²+ 36²
39²+ 10²= 1621 (prime) (ok)
39²+ 14²= 1717= 17* 101= 6²+ 41²
39²+ 16²= 1777 (prime) (ok)
39²+ 20²= 1921= 17* 113= 25²+ 36²
39²+ 22²= 2005= 5* 401= 18²+ 41²
39²+ 28²= 2305= 5* 461= 1²+ 48²
39²+ 32²= 2545= 5* 509= 12²+ 49²
39²+ 34²= 2677 (prime) (ok)
39²+ 38²= 2965= 5* 593= 7²+ 54²
41²+ 2²= 1685= 5* 337= 23²+ 34²
41²+ 4²= 1697 (prime) (ok)
41²+ 6²= 1717= 17* 101= 14²+ 39²
41²+ 8²= 1745= 5* 349= 28²+ 31²
41²+ 10²= 1781= 13* 137= 25²+ 34²
41²+ 12²= 1825= 5* 365= 15²+ 40²
41²+ 14²= 1877 (prime) (ok)
41²+ 16²= 1937= 13* 149= 1²+ 44²
41²+ 18²= 2005= 5* 401= 22²+ 39²
41²+ 20²= 2081 (prime) (ok)
41²+ 22²= 2165= 5* 433= 7²+ 46²
41²+ 24²= 2257= 37* 61= 31²+ 36²
41²+ 26²= 2357 (prime) (ok)
41²+ 28²= 2465= 5* 493= 8²+ 49²
41²+ 30²= 2581= 29* 89= 9²+ 50²
41²+ 32²= 2705= 5* 541= 1²+ 52²
41²+ 34²= 2837 (prime) (ok)
41²+ 36²= 2977= 13* 229= 24²+ 49²
41²+ 38²= 3125= 5* 625= 10²+ 55²
41²+ 40²= 3281= 17* 193= 16²+ 55²
43²+ 2²= 1853= 17* 109= 22²+ 37²
43²+ 4²= 1865= 5* 373= 29²+ 32²
43²+ 6²= 1885= 5* 377= 11²+ 42²
43²+ 8²= 1913 (prime) (ok)
43²+ 10²= 1949 (prime) (ok)
43²+ 12²= 1993 (prime) (ok)
43²+ 14²= 2045= 5* 409= 26²+ 37²
43²+ 16²= 2105= 5* 421= 13²+ 44²
43²+ 18²= 2173= 41* 53= 27²+ 38²
43²+ 20²= 2249= 13* 173= 32²+ 35²
43²+ 22²= 2333 (prime) (ok)
43²+ 24²= 2425= 5* 485= 11²+ 48²
43²+ 26²= 2525= 5* 505= 5²+ 50²
43²+ 28²= 2633 (prime) (ok)
43²+ 30²= 2749 (prime) (ok)
43²+ 32²= 2873= 13* 221= 8²+ 53²
43²+ 34²= 3005= 5* 601= 14²+ 53²
43²+ 36²= 3145= 5* 629= 3²+ 56²
43²+ 38²= 3293= 37* 89= 22²+ 53²
43²+ 40²= 3449 (prime) (ok)
43²+ 42²= 3613 (prime) (ok)
45²+ 2²= 2029 (prime) (ok)
45²+ 4²= 2041= 13* 157= 21²+ 40²
45²+ 8²= 2089 (prime) (ok)
45²+ 14²= 2221 (prime) (ok)
45²+ 16²= 2281 (prime) (ok)
45²+ 22²= 2509= 13* 193= 3²+ 50²
45²+ 26²= 2701= 37* 73= 10²+ 51²
45²+ 28²= 2809= 53* 53= 0²+ 53²
45²+ 32²= 3049 (prime) (ok)
45²+ 34²= 3181 (prime) (ok)
45²+ 38²= 3469 (prime) (ok)
45²+ 44²= 3961= 17* 233= 19²+ 60²
47²+ 2²= 2213 (prime) (ok)
47²+ 4²= 2225= 5* 445= 17²+ 44²
47²+ 6²= 2245= 5* 449= 33²+ 34²
47²+ 8²= 2273 (prime) (ok)
47²+ 10²= 2309 (prime) (ok)
47²+ 12²= 2353= 13* 181= 7²+ 48²
47²+ 14²= 2405= 5* 481= 2²+ 49²
47²+ 16²= 2465= 5* 493= 8²+ 49²
47²+ 18²= 2533= 17* 149= 33²+ 38²
47²+ 20²= 2609 (prime) (ok)
47²+ 22²= 2693 (prime) (ok)
47²+ 24²= 2785= 5* 557= 9²+ 52²
47²+ 26²= 2885= 5* 577= 22²+ 49²
47²+ 28²= 2993= 41* 73= 17²+ 52²
47²+ 30²= 3109 (prime) (ok)
47²+ 32²= 3233= 53* 61= 23²+ 52²
47²+ 34²= 3365= 5* 673= 1²+ 58²
47²+ 36²= 3505= 5* 701= 16²+ 57²
47²+ 38²= 3653= 13* 281= 17²+ 58²
47²+ 40²= 3809= 13* 293= 28²+ 55²
47²+ 42²= 3973= 29* 137= 2²+ 63²
47²+ 44²= 4145= 5* 829= 7²+ 64²
47²+ 46²= 4325= 5* 865= 10²+ 65²
49²+ 2²= 2405= 5* 481= 14²+ 47²
49²+ 4²= 2417 (prime) (ok)
49²+ 6²= 2437 (prime) (ok)
49²+ 8²= 2465= 5* 493= 16²+ 47²
49²+ 10²= 2501= 41* 61= 1²+ 50²
49²+ 12²= 2545= 5* 509= 32²+ 39²
49²+ 16²= 2657 (prime) (ok)
49²+ 18²= 2725= 5* 545= 15²+ 50²
49²+ 20²= 2801 (prime) (ok)
49²+ 22²= 2885= 5* 577= 26²+ 47²
49²+ 24²= 2977= 13* 229= 36²+ 41²
49²+ 26²= 3077= 17* 181= 31²+ 46²
49²+ 30²= 3301 (prime) (ok)
49²+ 32²= 3425= 5* 685= 17²+ 56²
49²+ 34²= 3557 (prime) (ok)
49²+ 36²= 3697 (prime) (ok)
49²+ 38²= 3845= 5* 769= 1²+ 62²
49²+ 40²= 4001 (prime) (ok)
49²+ 44²= 4337 (prime) (ok)
49²+ 46²= 4517 (prime) (ok)
49²+ 48²= 4705= 5* 941= 9²+ 68²
51²+ 2²= 2605= 5* 521= 29²+ 42²
51²+ 4²= 2617 (prime) (ok)
51²+ 8²= 2665= 5* 533= 19²+ 48²
51²+ 10²= 2701= 37* 73= 26²+ 45²
51²+ 14²= 2797 (prime) (ok)
51²+ 16²= 2857 (prime) (ok)
51²+ 20²= 3001 (prime) (ok)
51²+ 22²= 3085= 5* 617= 13²+ 54²
51²+ 26²= 3277= 29* 113= 19²+ 54²
51²+ 28²= 3385= 5* 677= 24²+ 53²
51²+ 32²= 3625= 5* 725= 5²+ 60²
51²+ 38²= 4045= 5* 809= 18²+ 61²
51²+ 40²= 4201 (prime) (ok)
51²+ 44²= 4537= 13* 349= 21²+ 64²
51²+ 46²= 4717= 53* 89= 19²+ 66²
51²+ 50²= 5101 (prime) (ok)
Thus the time has come to prove this conjecture mathematically... At first, we suppose that S is the sum of two "other" square numbers, and we prove that it comes from this that S is not prime. So our supposition is:
S=(2k)²+(2j+1)²=(2m)²+(2n+1)²
gcd(2k,2j+1)=1
First we convert the first equation to the following form:
(x-y)²+(z+v)²=(x+y)²+(z-v)²
where (z+v) is the greatest of the terms (2k), (2j+1), (2m) and (2n+1), and (z-v) is the other term which has the same parity as (z+v). This can be done, as (say) the following system of equations has one solution:
x-y=2k
z+v=2j+1
x+y=2m
z-v=2n+1
And it can be solved easily for x,y,z,v (they can be given in such a way that x>=1, z>=1,y>=1,v>=1, if -say- m>k and j>n):
x=k+m
z=j+n+1
y=m-k
v=j-n
Thus we can work with the variables x,y,z,v from now on. We can transform our equation in the following way:
(x-y)²+(z+v)²=(x+y)²+(z-v)²
x²+y²+z²+v²-2xy+2zv=x²+y²+z²+v²+2xy-2zv
4zv=4xy
zv=xy
... and using this identity, we can substitute it back into our equation, we get:
S=(x-y)²+(z+v)²=(x+y)²+(z-v)²=x²+y²+z²+v²
But it's still more interesting to us that T=z*v=x*y. In the prime factors of this product (T), some are element(s) of the set of factors of z, some are element(s) of the set of factors of v... and some are of x and some are of y. These four sets intersect in four subsets, which may be called a, b, c and d:
Thus we can write (say) the following system of equations:
x=ac
y=bd
z=ad
v=bc
Afterwards, we can square all equations the following way:
x²=a²c²
y²=b²d²
z²=a²d²
v²=b²c²
Adding them together gives:
S=x²+y²+z²+v²=a²c²+b²d²+a²d²+b²c²=(a²+b²)(c²+d²)=S
And we can reckon that this is equal to the sum S. As a, b, c and d are at least 1, S is a composite number, not prime. I've used the Brahmagupta-Fibonacci identity (otherwise called the Diophantus identity) when proving this part of the conjecture. (We can use the rule of contraposition to show that if S is prime, then it is not the sum of two other square numbers.) Note: the Brahmagupta-Fibonacci identity is this (can be proven by simple algebra):
(a²+b²)(c²+d²)=(ac-bd)²+(ad+bc)²=(ac+bd)²+(ad-bc)²
The other part of the conjecture seemed to be more tricky. This says that if S is not prime, then S is the sum of two other square numbers. Thus the following is supposed:
S=(2k)²+(2j+1)²=e*f
gcd(2k,2j+1)=1
e>1,f>1,k>=1,j>=0
... and we must prove that e*f is of the form:
e=a²+b²
f=c²+d²
If we managed to prove this, it would come from the "Brahmagupta-Fibonacci identity" that S can be expressed as the sums of two squares two ways, one of which may be different from (2k)²+(2j+1)², proving this part of the conjecture. Now what? :-) I've found some interesting theorems in WikiPedia that are related. These are the following:
Now what? :-) The "Sum of two squares theorem" says that:
An integer greater than one can be written as a sum of two squares if and only if its prime decomposition contains no prime congruent to 3 modulo 4 raised to an odd power.(https://en.wikipedia.org/wiki/Sum_of_two_squares_theorem#cite_note-1
Underwood Dudley (1978). Elementary Number Theory (2 ed.). W.H. Freeman and Company.)
Now S can be written as a sum of two squares, so in its prime decomposition all possible primes of the form (4p+3) are squared. Let the product of all of these be s². If s² is not a divisor of either e or f, then let's redefine e and f in such a way that e should be divisible by s² and f should also have some divisors of the form (4p+1). (If all that we have were only s², then S would be the sum of s² and 0², proving our conjecture, or if this were the same as our initial condition, then k>=1 would not hold.) According to "Fermat's theorem on sums of two squares", all the other primes of the form (4p+1) can be expressed as t²+u². According to the Brahmagupta-Fibonacci identity, when we multiply one such prime with another, the result can also be expressed as t²+u². In summary, this makes e of the form e=s²t²+s²u² and f=q²+r². Thus we have reached our desired goal that:
e=a²+b²
f=c²+d²
... and the Brahmagupta-Fibonacci identity proves the second part of the conjecture. (We can use the rule of contraposition to show that if S is not the sum of two other square numbers, then the number is prime.)
There is another interesting theorem worth mentioning here, Lagrange's four-square theorem. I hoped that it might help proving the Goldbach conjecture (with the result in this blog post), but it turned out not to be that case.
(Edited.) NOTE: The conjecture is not true when one of the square numbers is 0. I've discovered it when checking my post, and after modifying my C program I also get some counterexamples:
3²+ 0²= 9= 3* 3 (ok)
7²+ 0²= 49= 7* 7 (ok)
9²+ 0²= 81= 3* 27 (ok)
11²+ 0²= 121= 11* 11 (ok)
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