## Integer Factorization: 26 February 2023 Update and Screen Capture

What you see here is an attempt to look for patterns in the distribution of remainders that result from subtracting the first several smallest composite numbers from their Ceiling Squares, an ever-growing triangle of numbers made by taking ever-increasing values of s, with values of t less than s, relatively prime to s, even when s is odd, and odd when s is even, calculating m=s2-t2 from that, finding m’s Ceiling Square, and then producing the resultant remainder. I’ve placed the text file where I calculated all of these values right after, for reference’s sake. The top legend row of numbers is the values subtracted from the numbers running down the left side, the s values, to get t values. I did not start with 1 because s2-(s-1)2 suggests only the trivial factorization of odd integer 2s-1 for any s, and thus all odd integers, prime or composite.

Notice the diagonal-downward rows and double-rows of numbers. Study of these patterns, as always, seems like it could be fruitful. There are also weird things happening, marked “??” on the table and sometimes annotated in the text file that follows.

``````10^2-7^2 = 51 = 3*17, ascent 2, remainder 13.
10^2-3^2 = 91 = 7*13, ascent 0.
10^2-1^2 = 99 = 9*11, ascent 0.

11^2-8^2 = 57 = 3*19, ascent 2, remainder 24.
11^2-6^2 = 85 = 5*17, ascent 0.
11^2-4^2 = 105 = 7*15, ascent 0.
11^2-2^2 = 117 = 9*13, ascent 0.

12^2-7^2 = 95 = 5*19, ascent 2, remainder 5.
12^2-5^2 = 119 = 7*17, ascent 0.
12^2-1^2 = 143 = 11*13, ascent 0.

13^2-10^2 = 69 = 3*23, ascent 4, remainder 12.
13^2-8^2 = 105 = 7*15, ascent 0. (5*21 has ascent 2.)
13^2-6^2 = 133 = 7*19, ascent 0.
13^2-4^2 = 153 = 9*17, ascent 0.
13^2-2^2 = 165 = 11*15, ascent 0.

14^2-11^2 = 75 = 5*15, ascent 0. (3*25 has ascent 4.)
14^2-9^2 = 115 = 5*23, ascent 2, remainder 29.
14^2-5^2 = 171 = 9*19, ascent 0.
14^2-3^2 = 187 = 11*17, ascent 0.
14^2-1^2 = 195 = 13*15, ascent 0.

15^2-8^2 = 161 = 7*23, ascent 2, remainder 8.
15^2-4^2 = 209 = 11*19, ascent 0.
15^2-2^2 = 221 = 13*17, ascent 0.

16^2-13^2 = 87 = 3*29, ascent 6, remainder 13.
16^2-11^2 = 135 = 9*15, ascent 0. (5*27 has ascent 4.)
16^2-9^2 = 175 = 7*25, ascent 2, remainder 21.
16^2-7^2 = 207 = 9*23, ascent 0.
16^2-5^2 = 231 = 11*21, ascent 0.
16^2-3^2 = 247 = 13*19, ascent 0.
16^2-1^2 = 255 = 15*17, ascent 0.

17^2-14^2 = 93 = 3*31, ascent 6, remainder 28.
17^2-12^2 = 145 = 5*29, ascent 4, remainder 24.
17^2-10^2 = 189 = 9*21, ascent 0. (7*27 has ascent 2.)
17^2-8^2 = 225 = 15*15, ascent 0. (9*25 has ascent 2.)
17^2-6^2 = 253 = 11*23, ascent 0.
17^2-4^2 = 273 = 13*21, ascent 0.
17^2-2^2 = 285 = 15*19, ascent 0.

18^2-13^2 = 155 = 5*31, ascent 4, remainder 41.
18^2-11^2 = 203 = 7*29, ascent 2, remainder 53.
18^2-7^2 = 275 = 11*25, ascent 0.
18^2-5^2 = 299 = 13*23, ascent 0.
18^2-1^2 = 323 = 17*19, ascent 0.

19^2-16^2 = 105 = 7*15, ascent 0. (3*35 has ascent 8.)
19^2-14^2 = 165 = 11*15, ascent 0. (5*33 has ascent 6.)
19^2-12^2 = 217 = 7*31, ascent 4, remainder 8.
19^2-10^2 = 261 = 9*29, ascent 2, remainder 28.
19^2-8^2 = 297 = 11*27, ascent 0.
19^2-6^2 = 325 = 13*25, ascent 0.
19^2-4^2 = 345 = 15*23, ascent 0.
19^2-2^2 = 357 = 17*21, ascent 0.

20^2-17^2 = 111 = 3*37, ascent 8, remainder 33.
20^2-13^2 = 231 = 11*21, ascent 0. (7*33 has ascent 4.)
20^2-11^2 = 279 = 9*31, ascent 2, remainder 45.
20^2-9^2 = 319 = 11*29, ascent 2, remainder 5.
20^2-7^2 = 351 = 13*27, ascent 0.
20^2-3^2 = 391 = 17*23, ascent 0.
20^2-1^2 = 399 = 19*21, ascent 0.

21^2-16^2 = 185 = 5*37, ascent 6, remainder 40.
21^2-10^2 = 341 = 11*31, ascent 2, remainder 20.
21^2-8^2 = 377 = 13*29, ascent 0.
21^2-4^2 = 425 = 17*25, ascent 0.
21^2-2^2 = 437 = 19*23, ascent 0.

22^2-19^2 = 123 = 3*41, ascent 10, remainder 21.
22^2-17^2 = 195 = 13*15, ascent 0. (5*39 has ascent 8.)
22^2-15^2 = 259 = 7*37, ascent 4, remainder 65.
22^2-13^2 = 315 = 15*21, ascent 0. (9*35 has ascent 4.)
22^2-9^2 = 403 = 13*31, ascent 0.
22^2-7^2 = 435 = 15*29, ascent 0.
22^2-5^2 = 459 = 17*27, ascent 0.
22^2-3^2 = 475 = 19*25, ascent 0.
22^2-1^2 = 483 = 21*23, ascent 0.

23^2-20^2 = 129 = 3*43, ascent 10, remainder 40.
23^2-18^2 = 205 = 5*41, ascent 8, remainder 20.
23^2-16^2 = 273 = 13*21, ascent 0. (7*39 has ascent 6.)
23^2-14^2 = 333 = 9*37, ascent 4, remainder 28.
23^2-12^2 = 385 = 11*35, ascent 2, remainder 56.
23^2-10^2 = 429 = 13*33, ascent 2, remainder 12.
23^2-8^2 = 465 = 15*31, ascent 0.
23^2-6^2 = 493 = 17*29, ascent 0.
23^2-4^2 = 513 = 19*27, ascent 0.

24^2-19^2 = 215 = 5*43, ascent 8, remainder 41.
24^2-17^2 = 287 = 7*41, ascent 6, remainder 37.
24^2-13^2 = 407 = 11*37, ascent 2, remainder 77.
24^2-11^2 = 455 = 13*35, ascent 2, remainder 29.
24^2-7^2 = 527 = 17*31, ascent 0.
24^2-5^2 = 551 = 19*29, ascent 0.
24^2-1^2 = 575 = 23*25, ascent 0.

25^2-22^2 = 141 = 3*47, ascent 12, remainder 28.
25^2-18^2 = 301 = 7*43, ascent 6, remainder 60.
25^2-16^2 = 369 = 9*41, ascent 4, remainder 72.
25^2-14^2 = 429 = 13*33, ascent 2, remainder 10^2. (11*39 has ascent 4, remainder 12.) weird one!
25^2-12^2 = 481 = 13*37, ascent 2, remainder 48.
25^2-8^2 = 561 = 17*33, ascent 0.
25^2-6^2 = 589 = 19*31, ascent 0.
25^2-4^2 = 609 = 21*29, ascent 0.
25^2-2^2 = 621 = 23*27, ascent 0.

26^2-23^2 = 147 = 7*21, ascent 0. (3*49 has ascent 12.)
26^2-21^2 = 235 = 5*47, ascent 10, remainder 21.
26^2-19^2 = 315 = 15*21, ascent 0. (7*45 has ascent 8.)
26^2-17^2 = 387 = 9*43, ascent 6, remainder 13.
26^2-15^2 = 451 = 11*41, ascent 4, remainder 33.
26^2-11^2 = 555 = 15*37, ascent 2, remainder 21.
26^2-9^2 = 595 = 17*35, ascent 0.
26^2-7^2 = 627 = 19*33, ascent 0.
26^2-5^2 = 651 = 21*31, ascent 0.
26^2-3^2 = 667 = 23*29, ascent 0.
26^2-1^2 = 675 = 25*27, ascent 0.

27^2-22^2 = 245 = 7*35, ascent 4, remainder 284. (5*49 has ascent 10, remainder 44.) weird 2
27^2-20^2 = 329 = 7*47, ascent 8, remainder 32.
27^2-16^2 = 473 = 11*43, ascent 4, remainder 56.
27^2-14^2 = 533 = 13*41, ascent 2, remainder 92.
27^2-10^2 = 629 = 17*37, ascent 0.
27^2-8^2 = 665 = 19*35, ascent 0.
27^2-4^2 = 713 = 23*31, ascent 0.
27^2-2^2 = 725 = 25*29, ascent 0.

28^2-25^2 = 159 = 3*53, ascent 14, remainder 37.
28^2-23^2 = 255 = 15*17, aacent 0. (5*51 has ascent 12.)
28^2-19^2 = 423 = 9*47, ascent 6, remainder 61.
28^2-17^2 = 495 = 15*33, ascent 0. (11*45 has ascent 4.)
28^2-15^2 = 559 = 13*43, ascent 4, remainder 17.
28^2-13^2 = 615 = 15*41, ascent 2, remainder 61.
28^2-11^2 = 663 = 17*39, ascent 2, remainder 13.
28^2-9^2 = 703 = 19*37, ascent 0.
28^2-5^2 = 759 = 23*33, ascent 0.
28^2-3^2 = 775 = 25*31, ascent 0.
28^2-1^2 = 783 = 27*29, ascent 0.

29^2-26^2 = 165 = 11*15, ascent 0. (3*55 has ascent 16.)
29^2-24^2 = 265 = 5*53, ascent 12, remainder 24.
29^2-22^2 = 357 = 17*21, ascent 0. (7*51 has ascent 10.)
29^2-20^2 = 441 = 21*21, ascent 0. (9*49 has ascent 8.)
29^2-18^2 = 517 = 11*47, ascent 6, remainder 12.
29^2-16^2 = 585 = 15*39, ascent 2, remainder 144, (13*45 has ascent 4, remainder 40.) weird stuff in the ditches
29^2-14^2 = 645 = 15*43, ascent 2, remainder 84.
29^2-12^2 = 697 = 17*41, ascent 2, remainder 32.
29^2-10^2 = 741 = 19*39, ascent 0.
...

30^2-23^2 = 371 = 7*53, ascent 10, remainder 29.
30^2-19^2 = 539 = 11*49, ascent 6, remainder 37.
30^2-17^2 = 611 = 13*47, ascent 4, remainder 65.
30^2-13^2 = 731 = 17*43, ascent 2, remainder 53.
30^2-11^2 = 779 = 19*41, ascent 2, remainder 5.
30^2-7^2 = 851 = 23*37, ascent 0.
...

31^2-28^2 = 177 = 3*59, ascent 16, remainder 48.
31^2-26^2 = 285 = 15*19, ascent 0. (5*57 has ascent 14.)
31^2-24^2 = 385 = 11*35, ascent 2, remainder 456. (7*55 has ascent 10, remainder 56.)
31^2-22^2 = 477 = 9*53, ascent 8, remainder 52.
31^2-20^2 = 561 = 17*33, ascent 0. (11*51 has ascent 6.)
31^2-18^2 = 637 = 13*49, ascent 4, remainder 92.
31^2-16^2 = 705 = 15*47, ascent 4, remainder 24.
31^2-14^2 = 765 = 17*45, ascent 2, remainder 76.
31^2-12^2 = 817 = 19*43, ascent 2, remainder 24.
31^2-10^2 = 861 = 21*41, ascent 0.
...

32^2-29^2 = 183 = 3*61, ascent 18, remainder 13.
32^2-27^2 = 295 = 5*59, ascent 14, remainder 29.
32^2-25^2 = 399 = 19*21, ascent 0. (7*57 has ascent 12.)
32^2-23^2 = 495 = 15*33, ascent 0. (9*55 has ascent 8.)
32^2-21^2 = 583 = 11*53, ascent 6, remainder 93.
32^2-19^2 = 663 = 17*39, ascent 2, remainder 237. (13*51 has ascent 6, remainder 13.)
32^2-17^2 = 735 = 21*35, ascent 0. (15*49 has ascent 4.)
32^2-15^2 = 799 = 17*47, ascent 2, remainder 101.
32^2-13^2 = 855 = 19*45, ascent 2, remainder 45.
32^2-11^2 = 903 = 21*43, ascent 0.
...

33^2-28^2 = 305 = 5*61, ascent 14, remainder 56.
33^2-26^2 = 413 = 7*59, ascent 12, remainder 28.
33^2-20^2 = 689 = 13*53, ascent 6, remainder 40.
33^2-16^2 = 833 = 17*49, ascent 4, remainder 8.
33^2-14^2 = 893 = 19*47, ascent 2, remainder 68.
33^2-10^2 = 989 = 23*43, ascent 0.
...

34^2-31^2 = 195 = 13*15, ascent 0. (3*65 has ascent 20.)
34^2-29^2 = 315 = 15*21, ascent 0. (5*63 has ascent 16.)
34^2-27^2 = 427 = 7*61, ascent 12, remainder 57.
34^2-25^2 = 531 = 9*59, ascent 10, remainder 45.
34^2-23^2 = 627 = 19*33, ascent 0. (11*57 has ascent 8.)
34^2-21^2 = 715 = 13*55, ascent 6, remainder 69.
34^2-19^2 = 795 = 15*53, ascent 4, remainder 105.
34^2-15^2 = 931 = 19*49, ascent 2, remainder 93.
34^2-13^2 = 987 = 21*47, ascent 2, remainder 37.
34^2-11^2 = 1035 = 23*45, ascent 0.
...

35^2-32^2 = 201 = 3*67, ascent 20, remainder 24.
35^2-26^2 = 549 = 9*61, ascent 10, remainder 76.
35^2-24^2 = 649 = 11*59, ascent 8, remainder 80.
35^2-22^2 = 741 = 19*39, ascent 0. (13*57 has ascent 6.)
35^2-18^2 = 901 = 17*53, ascent 4, remainder 60.
35^2-16^2 = 969 = 19*51, ascent 2, remainder 120.
35^2-12^2 = 1081 = 23*47, ascent 2, remainder 8.
35^2-8^2 = 1161 = 27*43, ascent 0.
...

36^2-31^2 = 335 = 5*67, ascent 16, remainder 65.
36^2-29^2 = 455 = 13*35, ascent 2, remainder 701. (7*65 has ascent 14, remainder 29.)
36^2-25^2 = 671 = 11*61, ascent 10, remainder 5.
36^2-23^2 = 767 = 13*59, ascent 8, remainder 17.
36^2-19^2 = 935 = 17*55, ascent 4, remainder 89.
36^2-17^2 = 1007 = 19*53, ascent 4, remainder 17.
36^2-13^2 = 1127 = 23*49, ascent 2, remainder 29.
36^2-11^2 = 1175 = 25*47, ascent 0.
...
``````