[QUOTE=3.14159;233804]This statement can be distorted into some strange sort of crackpottery about some obscure pseudo-proof about some obscure pseudo-theorem.[/QUOTE]

I suppose any statement can, with enough (Levenshtein) effort...

Factors of 100-digit numbers;

8768935392562947618103320365401275447646525222169584791935566687049248245681935086698791112131179280 = 2[sup]4[/sup] * 5 * 7 * 229 * p95.

5013440241231690776250434282127412300350543560061260586054934713339836990739638416804096559649788785 = 5 * 13 * 9699973 * 1174300196009327 * 21278531935237641679 * 9306615253804741923704699323 * 34193173165859299637123433127.

Two can play that game.

1383597229331378476493295042730289164926892754069547350166289357488402239083316569155527338510512173 = 17 * 269 * 3204518441 * 697760349213711353909 * 5254579498151677786377319633 * 25751436085780192367390036356137118613

1226134517673268382787935537453090554039551405695194373164354832617671209370642217560662229657298470 = 2 * 3 * 5 * 131 * 619 * 628721 * 23852987 * 4724551547 * 10363456437839351504851859 * 686418473916303990455700991234048960669804471

4504635868071489691958247053230215472190113144381522888320546861068040654323199216497079846700145651 = 389 * 1249 * 9642973 * 2045233251289268647933076900071285541622713 * 470103901236160604912377322470525630823764859

1297098091109389427734623857619487484934106044886083070033690339492405858741869928141909886528666131 = 8317 * 11941 * 21327073 * 245809044309404623 * 59965612215340918627 * 41546460895888449456911011110597043412272105231

7448288592391672456339288547820229140957061588130168488233367000147015498831478910673540062648250669 = 7 * 23 * 61 * 827 * 7507 * 10477 * 19219 * 2307031 * 2497513 * 295393403176303499 * 69716460956822059780391 * 5112858994119557060987163901

2006581414691869533013736118005569210068797105484012314990944036018923405986462394666364748981293834 = 2 * 173 * 293 * 1069 * 4783 * 599093014896197532123744596093287948267 * 6461609735497599811892038514138831637953353686917

1029740705859616722557935599692863818295173314586300398254476931684298457572289392513244259443832990 = 2 * 5 * 73 * 199[SUP]2[/SUP] * 2803 * 17207646383 * 23341361566424419838977 * 108806050756253076134575189 * 290787051597685066876250670779

1479250575750629405583533546871938031514416077664003084605743742272187812384319917935052766003248180 = 2[SUP]2[/SUP] * 5 * 7 * 79 * 6871 * 25463 * 57047 * 61826203 * 358796701 * 799808381 * 1762119998489266483 * 428629288525583272511156276869590573727

2470685955216016137289558854053204746584489279198071269508371183297983597651454585258860897699629611 = 7 * 6229 * 346223 * 1077224275553 * 140671259709105592299571501927 * 1080024859552291303286190375729442638566243936449

2582787899792929109639761223985963200359419971336466309731940735382779345756386069144321580216562935 = 3[SUP]2[/SUP] * 5 * 7[SUP]2[/SUP] * 163 * 307 * 797 * 1723 * 71363 * 4865184453786808865917799950950769 * 49095101401206488590960022804247016107742250711

2491980943988512648154994168720542478864237721433866865571719881672143735588510153037196275622246583 = 3 * 37 * 1657 * 241489 * 303097 * 34848139 * 27108622699 * 85333941456585370142137 * 2296208868050293347620604354622609936964009

I have a challenge for you all;

k * n! + 1 = p[sup]2[/sup], where p is a prime integer, and where k ≤ n!

Ex: 5 * 8! + 1 = 449[sup]2[/sup]

Submissions: 256060*396^8560+1 (22242 digits)

Verification:

Primality testing 256060*396^8560+1 [N-1, Brillhart-Lehmer-Selfridge]
Running N-1 test using base 13
Special modular reduction using zero-padded FFT length 10K on 256060*396^8560+1
Calling Brillhart-Lehmer-Selfridge with factored part 40.08%
256060*396^8560+1 is prime! (15.0315s+0.0242s)

Next collection project...

-Work on the factorial + 1 primes.

If you want, you can get started on the -1 analogues.

No, not n! + 1; k * n! + 1.

Aaaaannnnddd, I'll kick off at.. 18, where PFGW cannot trivially factor them anymore.

You can dive for the smaller ones. However, I will extend the range by a factor of 1.5

Up to 107..

Covered every prime under 170 or so digits..

Now, I have covered every prime under 206 digits..

better than what I already know ?
 

if I did my math correct only certain k work for a given property of the exponent to try for factors I was wondering if we could every limit this down enough to make it easier.

[QUOTE=science_man_88;234256]if I did my math correct only certain k work for a given property of the exponent to try for factors I was wondering if we could every limit this down enough to make it easier.[/QUOTE]

What k do you find as impossible?

Okay; Back to the old game. Listing factors of 10[sup]200![/sup] -1.

I submit; 110742186470530054291318013, 10000099999999989999899999000000000100001, and 15362898429170396757717888856328974146292496901433891193564055671816191643.

Also; Is 990001 a long prime? It has no repetition for at least 1/2 its decimal expansion.

And lastly, is there a method to determine whether or not a certain prime is a long prime?

[QUOTE=CRGreathouse;234269]What k do you find as impossible?[/QUOTE]

well for example if p is 3 mod 8 then

for 2*k*p+1 to be = +1-1 mod 8

well 2*3 +1 = 7 so to be 7 mod 8 you have k=1

the next one that has mod 8 7 is k=5

and they follow k=4x+1

and if i did the math correct for 1 mod 8 it becomes

k=4x

If you're making a reference to the earlier challenge;

Find me a number of the form k * n! + 1 which is a prime square;

It's very easy. Here's an example: 5 * 8! + 1 = 449 * 449.

Note; Don't depend on anything relating to polynomials; They're all irreducible.

An example of a k * b^n + 1 number that is a prime power; 545793 * 396^3 + 1 = 5821793 * 5821793.