Mathematical Explorations: Maximally Composite

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``````