20180212, 21:54  #1 
"Rashid Naimi"
Oct 2015
Remote to Here/There
2197_{10} Posts 
Factoring n via factors of n1 or n+1
Are there any such algorithms?
Other than occasional hits of FLT primality test, that is. Thank you in advance Last fiddled with by a1call on 20180212 at 22:02 
20180212, 22:16  #2  
"Forget I exist"
Jul 2009
Dumbassville
20300_{8} Posts 
Quote:
Last fiddled with by science_man_88 on 20180212 at 22:18 

20180212, 22:22  #3 
"Rashid Naimi"
Oct 2015
Remote to Here/There
13^{3} Posts 
Very interesting concept, regardless.
But that's not what I had in mind. Are there any algorithms that use known factors of n1 or n+1 to find factor candidates of n? 
20180212, 22:35  #4 
Aug 2006
13533_{8} Posts 
It doesn't seem likely. I know all the factors of 2^57828929506774569941382167850941401232743376153559817438751 but that doesn't help me find factors of the number just below it.

20180212, 22:53  #5 
"Rashid Naimi"
Oct 2015
Remote to Here/There
13^{3} Posts 
How about special forms, say subtract a 1 from you example.

20180212, 22:57  #6 
"Forget I exist"
Jul 2009
Dumbassville
20C0_{16} Posts 

20180213, 02:22  #7 
"Rashid Naimi"
Oct 2015
Remote to Here/There
13^{3} Posts 
FWIW:
Much better and more complete implantations seem possible. Basic Conceptual Implementation of the fact that most small FermatPseudoprimes have factors which are 1 greater than integer multiples of 1 less than their smallest factor. Example: 341 = 11 x 31 = (10 + 1) x (3 x 10) +1 https://en.wikipedia.org/wiki/Fermat...Factorizations Code:
print("\n**** BZS100A by Rashid Naimi (a1call) ****"); print("Basic Conceptual Implementation of the fact that most small FermatPseudoprimes have factors which are 1 greater than integer multiples of 1 less than their smallest factor."); print("Example:\n341 = 11 x 31 = (10 + 1) x (3 x 10) +1"); print("https://en.wikipedia.org/wiki/Fermat_pseudoprime#Factorizations"); allocatemem(19^8); forstep(e=3,19^4,2,{ n=2^e1; if(isprime(n),next(1););\\Skip prime candidates o=n1; for(i=0,19^1,\\Increase this range for larger Exponents e \\print("i ",i); fordiv(o, g, \\ Only useful if all prime factors of n1 are known gg=2^i*g+1;\\print("gg ",gg); k= lift(Mod(n,gg)); if((k == 0) && ((gg) != n), print("\n***** 2^",e,"  1 is divisible by:";); print("**** ",g+1); next(3); ); ); ); print("\n***** 2^",e,"  1";); print("No factor found."); \\next(19); }) Code:
**** BZS100A by Rashid Naimi (a1call) **** Basic Conceptual Implementation of the fact that most small FermatPseudoprimes have factors which are 1 greater than integer multiples of 1 less than their smallest factor. Example: 341 = 11 x 31 = (10 + 1) x (3 x 10) +1 https://en.wikipedia.org/wiki/Fermat_pseudoprime#Factorizations ***** 2^9  1 is divisible by: **** 7 ***** 2^11  1 is divisible by: **** 23 ***** 2^15  1 is divisible by: **** 7 ***** 2^21  1 is divisible by: **** 7 ***** 2^23  1 is divisible by: **** 47 ***** 2^25  1 is divisible by: **** 31 ***** 2^27  1 is divisible by: **** 7 ***** 2^29  1 is divisible by: **** 59 ***** 2^33  1 is divisible by: **** 7 ***** 2^35  1 No factor found. ***** 2^37  1 is divisible by: **** 223 ***** 2^39  1 is divisible by: **** 7 ***** 2^41  1 No factor found. ***** 2^43  1 No factor found. ***** 2^45  1 is divisible by: **** 7 ***** 2^47  1 is divisible by: **** 283 ***** 2^49  1 is divisible by: **** 127 ***** 2^51  1 is divisible by: **** 7 ***** 2^53  1 is divisible by: **** 1591 ***** 2^55  1 No factor found. ***** 2^57  1 is divisible by: **** 7 ***** 2^59  1 No factor found. ***** 2^63  1 is divisible by: **** 7 ***** 2^65  1 is divisible by: **** 31 ***** 2^67  1 No factor found. ***** 2^69  1 is divisible by: **** 7 ***** 2^71  1 No factor found. ***** 2^73  1 is divisible by: **** 439 ***** 2^75  1 is divisible by: **** 7 ***** 2^77  1 No factor found. ***** 2^79  1 No factor found. ***** 2^81  1 is divisible by: **** 7 ***** 2^83  1 is divisible by: **** 167 ***** 2^85  1 is divisible by: **** 31 ***** 2^87  1 is divisible by: **** 7 ***** 2^91  1 is divisible by: **** 127 ***** 2^93  1 is divisible by: **** 7 ***** 2^95  1 No factor found. ***** 2^97  1 No factor found. ***** 2^99  1 is divisible by: **** 7 ***** 2^101  1 No factor found. ***** 2^103  1 No factor found. ***** 2^105  1 is divisible by: **** 7 ***** 2^109  1 No factor found. ***** 2^111  1 is divisible by: **** 7 ***** 2^113  1 is divisible by: **** 3391 ***** 2^115  1 No factor found. ***** 2^117  1 is divisible by: **** 7 ***** 2^119  1 No factor found. ***** 2^121  1 is divisible by: **** 23 ***** 2^123  1 is divisible by: **** 7 ***** 2^125  1 is divisible by: **** 31 ***** 2^129  1 is divisible by: **** 7 ***** 2^131  1 is divisible by: **** 263 ***** 2^133  1 is divisible by: **** 127 ***** 2^135  1 is divisible by: **** 7 ***** 2^137  1 No factor found. ***** 2^139  1 No factor found. ***** 2^141  1 is divisible by: **** 7 ***** 2^143  1 No factor found. ***** 2^145  1 is divisible by: **** 31 ***** 2^147  1 is divisible by: **** 7 ***** 2^149  1 No factor found. ***** 2^151  1 No factor found. ***** 2^153  1 is divisible by: **** 7 ***** 2^155  1 No factor found. ***** 2^157  1 No factor found. ***** 2^159  1 is divisible by: **** 7 ***** 2^161  1 No factor found. ***** 2^163  1 No factor found. ***** 2^165  1 is divisible by: **** 7 ***** 2^167  1 No factor found. ***** 2^169  1 is divisible by: **** 8191 ***** 2^171  1 is divisible by: **** 7 ***** 2^173  1 No factor found. ***** 2^175  1 is divisible by: **** 127 ***** 2^177  1 is divisible by: **** 7 ***** 2^179  1 is divisible by: **** 359 ***** 2^181  1 is divisible by: **** 5431 ***** 2^183  1 is divisible by: **** 7 ***** 2^185  1 is divisible by: **** 31 ***** 2^187  1 No factor found. ***** 2^189  1 is divisible by: **** 7 ***** 2^191  1 is divisible by: **** 383 ***** 2^193  1 No factor found. ***** 2^195  1 is divisible by: **** 7 ***** 2^197  1 No factor found. ***** 2^199  1 No factor found. ***** 2^201  1 is divisible by: **** 7 ***** 2^203  1 No factor found. ***** 2^205  1 is divisible by: **** 31 ***** 2^207  1 is divisible by: **** 7 ***** 2^209  1 No factor found. ***** 2^211  1 is divisible by: **** 3799 ***** 2^213  1 is divisible by: **** 7 ***** 2^215  1 No factor found. ***** 2^217  1 is divisible by: **** 127 ***** 2^219  1 is divisible by: **** 7 ***** 2^221  1 No factor found. ***** 2^223  1 No factor found. ***** 2^225  1 is divisible by: **** 7 ***** 2^227  1 No factor found. ***** 2^229  1 No factor found. ***** 2^231  1 is divisible by: **** 7 ***** 2^233  1 is divisible by: **** 1399 ***** 2^235  1 No factor found. ***** 2^237  1 is divisible by: **** 7 ***** 2^239  1 is divisible by: **** 479 ***** 2^241  1 No factor found. ***** 2^243  1 is divisible by: **** 7 ***** 2^245  1 is divisible by: **** 31 ***** 2^247  1 No factor found. ***** 2^249  1 is divisible by: **** 7 ***** 2^251  1 is divisible by: **** 503 ***** 2^253  1 is divisible by: **** 271 ***** 2^255  1 is divisible by: **** 7 ***** 2^257  1 No factor found. ***** 2^259  1 is divisible by: **** 127 ***** 2^261  1 is divisible by: **** 7 
20180213, 02:40  #8 
"Sam"
Nov 2016
2^{2}·83 Posts 
The exponent, 57828929506774569941382167850941401232743376153559817438751 is prime, so you know that each prime dividing 2^578289295067745699413821678509414012327433761535598174387511 is congruent to 1 (mod 57828929506774569941382167850941401232743376153559817438751). That's useful because now you that 2^578289295067745699413821678509414012327433761535598174387511 has no prime factors below 5.782892e+58...

20180213, 02:49  #9  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:


20180213, 03:10  #10 
"Rashid Naimi"
Oct 2015
Remote to Here/There
13^{3} Posts 
Again FWIW:
the concept is based on the observation that just about all small FermatPseudoprimes have factors of the form: n = (x+1)(xy+1) ... The pattern gets more complex for larger numbers but the basic characteristic is usually still present, But higher orders of some of the prime factors of the smallest factor combine to produce the other factors. In it's basic form: n = (x+1)(xy+1) = x2y+x+xy+1 = x(xy+1+y)+1 => x  n1 Which is the link between the factors of n1 and the factors of n. This is a Theoretical concept and the application is still tentative to the caveat that all of the prime factors of n1 need to be known, This is often not the case with Merssene Pseudoprimes. So it does not have a practical application for factoring Merssene Pseudoprimes. Last fiddled with by a1call on 20180213 at 03:30 
20180213, 03:45  #11  
"Forget I exist"
Jul 2009
Dumbassville
8384_{10} Posts 
Quote:


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