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It’s time for some fun and games with some square-free “maximally composite” odd numbers and all their representations as the difference of perfect squares.
It’s kind of like following a growing pristine wave with patterns I didn’t expect to encounter, until it smashes into foam and chaos against a rocky shore.
15 = 8^2 - 7^2 (prime squared)
= 4^2 - 1^2 (unity squared).
105 = 53^2 (prime squared) - 52^2
= 19^2 ( " " ) - 16^2
= 13^2 ( " " ) - 8^2
= 11^2 ( " " ) - 4^2.
1155 = 578^2 - 577^2 (prime squared)
= 194^2 - 191^2 ( " " )
= 118^2 - 113^2 ( " " )
= 86^2 - 79^2 ( " " )
= 58^2 - 47^2 ( " " )
= 46^2 - 31^2 ( " " )
= 38^2 - 17^2 ( " " )
= 34^2 - 1^2 (unity squared).
15015 = 7508^2 - 7507^2 (prime squared)
= 2504^2 - 2501^2 (41^2*61^2)
= 1504^2 - 1499^2 (prime squared)
= 1076^2 - 1069^2 ( " " )
= 688^2 - 677^2 ( " " )
= 584^2 - 571^2 ( " " )
= 508^2 - 493^2 (17^2*29^2)
= 368^2 - 347^2 (prime squared)
= 244^2 - 211^2 ( " " )
= 232^2 - 197^2 ( " " )
= 212^2 - 173^2 ( " " )
= 164^2 - 109^2 ( " " )
= 148^2 - 83^2 ( " " )
= 136^2 - 59^2 ( " " )
= 128^2 - 37^2 ( " " )
= 124^2 - 19^2 ( " " ).
255255 = 127628^2 - 127627^2 (23^2*31^2*179^2)
= 42544^2 - 42541^2 (19^2*2239^2)
= 25528^2 - 25523^2 (prime squared)
= 18236^2 - 18229^2 ( " " )
= 11608^2 - 11597^2 ( " " )
= 9824^2 - 9811^2 ( " " )
= 8516^2 - 8501^2 ( " " )
= 7516^2 - 7499^2 ( " " )
= 6088^2 - 6067^2 ( " " ) 21 * 12155
= 3884^2 - 3851^2 ( " " ) 33 * 7735
= 3664^2 - 3629^2 (19^2*191^2) 35 * 7293
= 3292^2 - 3253^2 (prime squared) 39 * 6545
= 2528^2 - 2477^2 ( " " ) 51 * 5005
= 2348^2 - 2293^2 ( " " ) 55 * 4641
= 1996^2 - 1931^2 ( " " ) 65 * 3927
= 1696^2 - 1619^2 ( " " ) 77 * 3315
= 1544^2 - 1459^2 ( " " ) 85 * 3003
= 1448^2 - 1357^2 (23^2*59^2) 91 * 2805
= 1268^2 - 1163^2 (prime squared) 105 * 2431
= 1132^2 - 1013^2 ( " " ) 119 * 2145 (I love how 119 and 1132 are here!)
= 964^2 - 821^2 ( " " ) 143 * 1785
= 856^2 - 691^2 ( " " ) 165 * 1547
= 776^2 - 589^2 (19^2*31^2) 187 * 1365
= 752^2 - 557^2 (prime squared) 195 * 1309
= 688^2 - 467^2 ( " " ) 221 * 1155
= 668^2 - 437^2 (19^2*23^2) 231 * 1105
= 628^2 - 373^2 (prime squared) 255 * 1001
= 604^2 - 331^2 ( " " ) 273 * 935
= 536^2 - 179^2 ( " " ) 357 * 715
= 524^2 - 139^2 ( " " ) 385 * 663
= 512^2 - 83^2 ( " " ) 429 * 595
= 508^2 - 53^2 ( " " ). 455 * 561
4849845 = 2424923^2 (109^2*22247^2) - 2424922^2 1 * 4849845
= 808309^2 (prime squared) - 808306^2 3 * 1616615
= 484987^2 ( " " ) - 484982^2 5 * 969969
= 346421^2 ( " " ) - 346414^2 7 * 692835
= 220453^2 (89^2*2477^2) - 220442^2 11 * 440895
= 186539^2 (167^2*1117^2) - 186526^2 13 * 373065
= 161669^2 (269^2*601^2) - 161654^2 15 * 323323
= 142651^2 (29^2*4919^2) - 142634^2 17 * 285285
= 127637^2 (prime squared) - 127618^2 19 * 255255
= 115483^2 (23^2*5021^2) - 115462^2 21 * 230945
= 73499^2 (67^2*1097^2) - 73466^2 33 * 146965
= 69301^2 (37^2*1873^2) - 69266^2 35 * 138567
= 62197^2 (37^2*1681^2) - 62158^2 39 * 124355
= 47573^2 (113^2*421^2) - 47522^2 51 * 95095
= 44117^2 (157^2*281^2) - 44062^2 55 * 88179
= 42571^2 (prime squared) - 42514^2 57 * 85085
= 37339^2 ( " " ) - 37274^2 65 * 74613
= 31531^2 ( " " ) - 31454^2 77 * 62985
= 28571^2 ( " " ) - 28486^2 85 * 57057
= 26693^2 ( " " ) - 26602^2 91 * 53295
= 25573^2 (107^2*239^2) - 25478^2 95 * 51051
= 23147^2 (79^2*293^2) - 23042^2 105 * 46189
= 20437^2 (107^2*191^2) - 20318^2 119 * 40755
= 18299^2 (29^2*631^2) - 18166^2 133 * 36465
= 17029^2 (prime squared) - 16886^2 143 * 33915
= 14779^2 ( " " ) - 14614^2 165 * 29393
= 13061^2 (37^2*353^2) - 12874^2 187 * 25935
= 12533^2 (83^2*151^2) - 12338^2 195 * 24871
= 11707^2 (23^2*509^2) - 11498^2 209 * 23205
= 11083^2 (prime squared) - 10862^2 221 * 21945
= 10613^2 ( " " ) - 10382^2 231 * 20995
= 9941^2 ( " " ) - 9694^2 247 * 19635
= 9637^2 (23^2*419^2) - 9382^2 255 * 19019
= 9019^2 (29^2*311^2) - 8746^2 273 * 17765
= 8651^2 (41^2*211^2) - 8366^2 285 * 17017
= 7669^2 (prime squared) - 7346^2 323 * 15015
= 6971^2 ( " " ) - 6614^2 357 * 13585
= 6491^2 ( " " ) - 6106^2 385 * 12597
= 6277^2 ( " " ) - 5878^2 399 * 12155
= 5867^2 ( " " ) - 5438^2 429 * 11305
= 5557^2 ( " " ) - 5102^2 455 * 10659
= 4603^2 ( " " ) - 4042^2 561 * 8645
= 4373^2 ( " " ) - 3778^2 595 * 8151
= 4181^2 (37^2*113^2) - 3554^2 627 * 7735
= 3989^2 (prime squared) - 3326^2 663 * 7315
= 3979^2 (23^2*173^2) - 3314^2 665 * 7293
= 3749^2 (23^2*163^2) - 3034^2 715 * 6783
= 3643^2 (prime squared) - 2902^2 741 * 6545
= 3061^2 ( " " ) - 2126^2 935 * 5187
= 2987^2 (29^2*103^2) - 2018^2 969 * 5005
= 2923^2 (37^2*79^2) - 1922^2 1001 * 4845
= 2843^2 (prime squared) - 1798^2 1045 * 4641
= 2747^2 (41^2*67^2) - 1642^2 1105 * 4389
= 2677^2 (prime squared) - 1522^2 1155 * 4199
= 2581^2 (29^2*89^2) - 1346^2 1235 * 3927
= 2507^2 (23^2*109^2) - 1198^2 1309 * 3705
= 2459^2 (prime squared) - 1094^2 1365 * 3553
= 2389^2 ( " " ) - 926^2 1463 * 3315
= 2341^2 ( " " ) - 794^2 1547 * 3135
= 2309^2 ( " " ) - 694^2 1615 * 3003
= 2267^2 ( " " ) - 538^2 1729 * 2805
= 2251^2 ( " " ) - 466^2 1785 * 2717
= 2213^2 ( " " ) - 218^2 1995 * 2431
= 2203^2 ( " " ) - 58^2. 2145 * 2261
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