[QUOTE=CRGreathouse;233786]No. There are 2319 primes under 44,520 which divide 10[SUP]200![/SUP] - 1 and 2308 which do not. The latter list begins:

[code]2,5,467,479,503,563,587,719,839,863,887,983,1019,1109,1187,1229,1283,1307,1319,1367,1399,1439,1447,1487,1493,1523,1543,1579,1619,1627,1637,1663,1699,1733,1759,1823,1867,1879,1907,1913,1949,1987,1997,2027,2039,2063,2083,2099,2111,2153,2203,2207,2239,2309,2383,2411,2447,2459,2477,2579,2659,2677,2693,2711,2749,2767,2777,2797,2803,2819,2837,2879,2903,2909,2957,2963,2999,3019,3023,3119,3167,3203,3209,3229,3253,3343,3347,3371,3373,3413,3449,3463,3467,3491,3517,3533,3559,3593,3607,3623,3643,3671,3677,3733,3767,3779,3803,3833,3847,3863,3919,3929,3947,3967,3989,4007,4073,4079,4091,4099,4127,4133,4139,4157,4211,4253,4259,4283,4337,4339,4349,4363,4373,4391,4457,4493,4507,4517,4519,4547,4549,4567,4597,4639,4643,4679,4703,4723,4783,4787,4793,4799,4813,4871,4909,4919,4937,4943,4987,5009,5021,5039,5059,5087,5099,5119,5179,5189,5197,5231,5261,5273,5297,5303,5309,5323,5381,5387,5393,5399,5417,5443,5449,5471,5483,5507,5527,5557,5623,5639,5647,5683,5693,5711,5717,5737,5749,5807,5813,5827,5861,5867,5879,5903,5927,5939,6011,6037,6047,6067,6079,6089,6131,6173,6197,6199,6203,6221,6247,6269,6277,6287,6311,6317,6353,6367,6379,6389,6473,6547,6569,6599,6607,6619,6653,6659,6691,6703,6719,6737,6779,6793,6823,6827,6829,6857,6871,6899,6907,6911,6983,6991,7013,7027,7043,7079,7109,7127,7159,7187,7207,7213,7219,7247,7283,7307,7331,7369,7433,7457,7517,7523,7529,7559,7573,7577,7583,7589,7607,7643,7649,7691,7699,7703,7717,7727,7757,7759,7793,7817,7823,7883,7927,7933,7949,7963,8039,8069,8087,8089,8111,8117,8147,8167,8219,8231,8243,8287,8291,8293,8311,8329,8369,8377,8387,8389,8423,8431,8539,8543,8563,8573,8597,8599,8609,8623,8627,8629,8677,8699,8707,8719,8747,8753,8783,8819,8831,8839,8861,8863,8887,8923,8963,8999,9013,9059,9067,9133,9137,9161,9173,9187,9209,9221,9227,9277,9319,9337,9341,9371,9377,9391,9403,9419,9437,9467,9479,9497,9511,9533,9539,9587,9619,9623,9643,9679,9733,9739,9743,9749,9767,9787,9817,9833,9839,9887,9931,9949,9973,...[/code][/QUOTE]

How is it that the primes above do not divide 10[sup]200![/sup] - 1?

1/503 has a period of 2*251...

251 does not divide 200!.

[QUOTE=3.14159;233787]How is it that the primes above do not divide 10[sup]200![/sup] - 1?[/QUOTE]

The number is 1 mod 2 and 4 mod 5, so those are obvious.

For the others, I'm suddenly unsure... I need to check my code.

Correct on 1019, 1109, 1187, 1229, 1283, 1307, 1319, ..

In general, long primes that are the larger analogues of SG primes shouldn't divide the number. For all other cases, they indeed divide the number.

Large factor;

[code]212158877458214105082975156638861697015962263232575666670500377582336595545516027747130897531745104763279308963402608489579488948174141288057321985065300709739123779635351802073368031960173399741862474936796650491652633364677473730132427004187[/code]

[QUOTE=3.14159;233789]Correct on 1019, 1109, 1187, 1229, 1283, 1307, 1319, ..

In general, long primes that are the larger analogues of SG primes shouldn't divide the number. For all other cases, they indeed divide the number.[/QUOTE]

I get 10[SUP]200![/SUP] - 84 is divisible by 467, and thus 10[SUP]200![/SUP] - 1 is not. Is my bignum library failing me?

Edit: I don't think so. 200! is 214 mod phi(467) and so 10[sup]200![/sup] mod 467 is 10[sup]214[/sup] mod 467, which I can solve by hand easily enough.

467 indeed doesn't divide the number. The period for 1/467 is 233. 233 does not divide 200!, therefore 467 does not divide 10[sup]200![/sup]-1.

So which, if any, of my numbers do you dispute?

[QUOTE=3.14159;233782]But still, any prime under 44520 divides 10[sup]200![/sup] -1.[/QUOTE]

2? 5?

Any primitive factors? (that is, factors that do not divide algebraic factors.)

My last small Cullen/Woodall from that same sieving batch.

[code]659*402^659+1
406*404^406+1
658*411^658-1
605*412^605-1
605*416^605+1
698*418^698-1
469*420^469+1
538*420^538+1
524*421^524-1
600*425^600-1
686*429^686+1
524*430^524-1
668*431^668-1
490*438^490-1
558*438^558-1
667*440^667-1
680*445^680-1
469*446^469-1
504*447^504-1
561*456^561+1
609*456^609-1
678*466^678-1
552*475^552-1
572*476^572-1
503*480^503-1
577*480^577+1
601*480^601-1
525*482^525+1
605*484^605-1
691*488^691-1
506*489^506+1
590*492^590+1
588*498^588+1
693*524^693+1
624*525^624-1
591*536^591+1
646*552^646-1
560*559^560-1
579*564^579+1
638*572^638-1
638*574^638-1
591*580^591+1
694*598^694+1
692*611^692-1
631*618^631-1
700*618^700-1
621*620^621+1
684*626^684-1
699*630^699-1
648*638^648-1
687*640^687-1
693*678^693-1[/code]

More small ones... and some bigger

[code]6*400^400-1
10*455^455-1
5*628^628-1
10*738^738-1
4*885^885-1
7*1128^1128-1
2*1400^1400-1[/code]

Special modular reduction using all-complex FFT length 192 on 2^3454+7
2^3454+7 is 3-PRP! (4.9997s+0.0004s)
Special modular reduction using all-complex FFT length 192 on 2^3510+7
2^3510+7 is 3-PRP! (0.0194s+0.0010s)
Generic modular reduction using generic reduction FFT length 384 on A 3572-bit number
((12^60+1)^2-2)*(11^908-1)+1 is 3-PRP! (14.9898s+0.0013s)
Special modular reduction using all-complex FFT length 192 on 2^3613+5
(2^3613+5)/7 is 3-PRP! (4.9933s+0.0021s)
Special modular reduction using all-complex FFT length 192 on 2^3749+9
2^3749+9 is 3-PRP! (0.0202s+0.0009s)
Special modular reduction using all-complex FFT length 192 on 2^3868+5
(2^3868+5)/21 is 3-PRP! (4.9862s+0.0012s)
Special modular reduction using all-complex FFT length 256 on 2^3864+7
2^3864+7 is 3-PRP! (4.9858s+0.0010s)
Special modular reduction using all-complex FFT length 256 on 2^3870+7
2^3870+7 is 3-PRP! (0.0273s+0.0008s)
Generic modular reduction using generic reduction FFT length 384 on A 3939-bit number
((12^60+1)^2-2)*(11^1014-1)+1 is 3-PRP! (14.9960s+0.0012s)
Generic modular reduction using generic reduction FFT length 448 on A 4886-bit number
(((12^60+1)^2-2)*(7^1588+1)+1)/5 is 3-PRP! (25.0065s+0.0012s)

[QUOTE=lorgix;233826][code]6*400^400-1
10*455^455-1
5*628^628-1
10*738^738-1
4*885^885-1
7*1128^1128-1
2*1400^1400-1[/code][/QUOTE]

I've not yet displayed the -1 side of k*b^b+/-1, but all primes for k<=10000 and b<=1000 are known. Perhaps I can add those today.

The +1 side is avalaible [url=www.rieselprime.de/Others/kbbp.htm]here[/url].