## Integer Factorization FYI: Current “scratch paper” worksheet

Copied from my text editor app, just so show lots of recent cogitation evidence.

``````a^2-b^2=c^2-d^2
a^2+d^2=b^2+c^2
a^2-c^2=b^2-d^2

(a+b)(a-b)=(c+d)(c-d)
(a+c)(a-c)=(b+d)(b-d)

8^2-7^2=4^2-1^2, factoring 15 two ways
8^2+1^2=4^2+7^2
8^2-4^2=7^2-1^2, factoring 48 two ways

11^2-10^2=5^2-2^2, factoring 21 two ways
11^2+2^2=5^2+10^2
11^2-5^2=10^2-2^2, factoring 96 = 48*2 two ways

17^2-16^2=7^2-4^2, factoring 33 two ways
17^2+4^2=7^2+16^2
17^2-7^2=16^2-4^2, factoring 240 = 48*5 two ways

18^2-17^2=6^2-1^2, factoring 35 two ways
18^2+1^2=6^2+17^2
18^2-6^2=17^2-1^2, factoring 288 = 48*6 two ways

20^2-19^2=8^2-5^2, factoring 39 two ways
20^2+5^2=8^2+19^2
20^2-8^2=19^2-5^2, factoring 336 = 48*7 two ways

28^2-27^2=8^2-3^2, factoring 55 two ways
28^2+3^2=8^2+27^2
28^2-8^2=27^2-3^2, factoring 720 = 48*15 two ways

39^2-38^2=9^2-2^2, factoring 77 two ways
39^2+2^2=9^2+38^2
39^2-9^2=38^2-2^2, factoring 1440 = 48*30 two ways

m=pq.
((m+1)/2)^2-((m-1)/2)^2=((p+q)/2)^2-((p-q)/2)^2, factoring m two ways
((m+1)/2)^2-((p+q)/2)^2=((m-1)/2)^2-((p-q)/2)^2, resulting in
(m+p+q+1)(m-p-q+1)=(m+p-q-1)(m-p+q-1)

1313 = 37^2-(8^2-(3^2-1^2))		= 37^2  -56 = 44^2-(40*47) = 656^2-(310*8*347)
= 39^2-(15^2-(5^2-(3^2-1^2)))	= 39^2 -208
= 41^2-(20^2-(6^2-2^2))		= 41^2 -368
= 43^2-(24^2-(7^2-3^2))		= 43^2 -536
= 45^2-(27^2-(5^2-(3^2-1^2)))	= 45^2 -712 (44^2-1224) (24*51)
= 47^2-(30^2-2^2)			= 47^2 -896 (44^2-1040) (20*52)
= 49^2-(33^2-1^2)			= 49^2-1088 (44^2-848)  (16*53)
= 51^2-(36^2-(3^2-1^2)		= 51^2-1288 (44^2-648)  (12*54)
= 53^2-(39^2-5^2)			= 53^2-1496 (44^2-440)   (8*55)
= 55^2-(42^2-(8^2-(4^2-2^2)))	= 55^2-1712 (44^2-224) (1204*356)
= 57^2-44^2			= 57^2-1936 ... (656^2-428400) (1200*357)

= 653^2-425096 (656^2-5240) (8*655)
= 655^2-427712 (656^2-2624) (4*656)
= 657^2-656^2			= 657^2-430336.

37^2-56 = 657^2-656^2.
37^2+656^2 = 657^2+56.
37^2-657^2 = 56-656^2.
657^2-37^2 = 656^2-56.

57^2-44^2 = 657^2-656^2.
57^2+656^2 = 44^2+657^2.
657^2-57^2 = 656^2-44^2.
600 * 714 = 700 * 612.

[long list of iterations of series of ceiling squares deleted]

4867 = 70^2-33        (33=2433^2-5919456)    (5919456=4728*1252} 2^3*3*197    * 2^2*313
= 72^2-317       (317=2433^2-5919172)   (5919172=4724*1253) 2^2*1181     * 7*179
= 74^2-609       (609=2433^2-5918880)   (5918880=4720*1254) 2^4*5*59     * 2*3*11*19
4716*1255  2^2*3^2*131  * 5*251
4712*1256  2^3*19*31    * 2^3*157
4708*1257  2^2*11*107   * 3*419
...
= 94^2-63^2      (63^2=2433^2-5915520)  (5915520=4680*1264) 2^3*3^2*5*13 * 2^4*79
(60*78)
...
= 2430^2-5900033 (5900033=2433^2-19456) (19556=8*2432)
= 2432^2-5909757 (5909757=2433^2-9732)  (9732=4*2433)
= 2434^2-2433^2

3936=2^5*3*41.
5919456=2^5*3*197*313

3969 = 63^2.
3782 = 63^2-187 = 62^2-62.
3597 = 63^2-372 = 62^2-247 = 61^2-124 = 60^2-3.
...

33  = 25^2-592 = 19^2-328 = 7^2-16.
317 = 25^2-308 = 19^2-44  = 7^2+268.
609 = 25^2-16  = 19^2+248 = 7^2+560.

12403 = 112^2 - 141.
12403 = 114^2 - 593.
12403 = 116^2 - 1053.
12403 = 118^2 - 1521.

141  = 13^2-28   = 25^2-484 = 33^2-948 = 39^2-1380.
593  = 13^2+424  = 25^2-32  = 33^2-496 = 39^2-928.
1053 = 13^2+884  = 25^2+428 = 33^2-36  = 39^2-468.
1521 = 13^2+1352 = 25^2+896 = 33^2+432 = 39^2.

2^2*7    2^2*121  2^2*237 2^2*345 (3*5*23)
-2^2*106  2^2*8    2^2*124 2^2*232 (2^3*29)
-2^2*221 -2^2*107  2^2*9   2^2*117 (3^2*13)
-2^2*338 -2^2*224 -2^2*108 0

2^2-0^2 = 4. n = 1.
3^2-1^2 = 8. n = 2.
4^2-2^2 = 12. n = 3.

(n-2)^2 = n^2-4n+4.
n^2-(n-2)^2 = 4n-4 = 4(n-1).

(n-2a)^2 = n^2-4an+4a^2.
n^2-(n-2a)^2 = 4an-4a^2 = 4a(n-a).

5809 = 77^2 - 120.
5809 = 79^2 - 432.
5809 = 81^2 - 752.

120 = 11^2-1   = 21^2-321 = 29^2-721.
432 = 11^2+311 = 21^2-9   = 29^2-409.
752 = 11^2+631 = 21^2+311 = 29^2-89.

diffs are 312, 320, 328, ...
0
312 = 2^3*3*13
320 = 2^6*5

101010101 = 10101^2-1010^2.

TL skew numbers: -236, -756, -852.    2^2*59 2^2*3^3*7 2^2*3*71
BR skew numbers: -44, -192, 712, 808. 2^2*11 2^6*3 2^3*89 2^3*101

f(x) = x^2+2cx+r = (x+c)^2-m.

(x+c)^2-m = a^2.
(x+c)^2 = a^2+m.
x+c = sqrt(a^2+m).
x = sqrt(a^2+m)-c.

m = c^2 - r
= (c+2)^2 - ( r + 4c + 4 )
= (c+n)^2 - ( r + 2nc + n^2 )

In the case of 5809,

m = (c+20)^2 - (120 + 40c + 400).

2^3 * 3 * 5
2^4 * 3^3
2^4 * 47
2^3 * 3^3 * 5
2^3 * 3 * 59
2^5 * 5 * 11
2^6 * 3 * 11
2^3 * 3 * 103
2^3 * 5 * 71
2^4 * 3 * 67
2^4 * 3^2 * 5^2

r + 2nc + n^2 = a^2 for some a.

(n+1)^2-(n-1)^2 = n^2 + 2n + 1 - n^2 + 2n - 1 = 4n.

x=sqrt(a^2+m)-c.
x=sqrt(a^2+11882081)-3449.

51 = 10^2-7^2 = 3*17
=  8^2-13 (rem diff 36).

57 = 11^2-8^2 = 3*19
=  9^2-24 (rem diff 40).

95 = 12^2-7^2 = 5*19
= 10^2-5 (rem diff 44).

69 = 13^2-10^2 = 3*23
= 11^2-52 (rem diff 48) = 9^2-12 (rem diff 88).

115 = 14^2-9^2 = 5*23
= 12^2-29 (rem diff 52).

161 = 15^2-8^2 = 7*23
= 13^2-8 (rem diff 56).

87 = 16^2-13^2 = 3*29
= 14^2-109 (rem diff 60) = 10^2-13 (rem diff 156).

145 = 17^2-12^2 = 5*29
= 15^2-80 (rem diff 64) = 13^2-24 (rem diff 120).

203 = 18^2-11^2 = 7*29
= 16^2-53 (rem diff 68).

319 = 20^2-9^2 = 11*29
= 18^2-5 (rem diff 76).

93 = 17^2-14^2 = 3*31
= 15^2-132 (rem diff 64) = 11^2-28 (rem diff 168).

155 = 18^2-13^2 = 5*31
= 16^2-101 (rem diff 68) = 14^2-41 (rem diff 128).

217 = 19^2-12^2 = 7*31
= 17^2-72 (rem diff 72) = 15^2-8 (rem diff 136).

341 = 21^2-10^2 = 11*31
= 19^2-20 (rem diff 80).

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