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-   -   Further replies to carpetpool's generalized pierpont prime thread (https://www.mersenneforum.org/showthread.php?t=23075)

science_man_88 2018-02-21 03:02

Yes, my code from [URL="http://www.mersenneforum.org/showthread.php?t=23069"]the thread[/URL] has a slight typo, should say, y-1. But my point is primes with ⁴√-1 possible for all the bases may have 33+ forms divisible by them. Eliminating a lot of forms right off the start. For 17 these include:

[CODE]
2^(8x)*3^(16y+8)*5^(16z+8)*7^(16a+8)+1
2^(8x+4)*3^(16y)*5^(16z+8)*7^(16a+8)+1
2^(8x+4)*3^(16y+8)*5^(16z)*7^(16a+8)+1
2^(8x+4)*3^(16y+8)*5^(16z+8)*7^(16a)+1
2^(8x)*3^(16y)*5^(16z)*7^(16a+8)+1
2^(8x)*3^(16y)*5^(16z+8)*7^(16a)+1
2^(8x)*3^(16y+8)*5^(16z)*7^(16a)+1
2^(8x+4)*3^(16y)*5^(16z)*7^(16a)+1
forprime(x=1,10,forprime(y=x+1,100, if(y%4==1,for(z=1,y-1,if(lift(Mod(x,y)^z)==sqrt(Mod(-1,y)),print(x","y","z);next(2))))))
for all the sqrt of -1 cases( except PARI only knows one of them). And I think 1 mod 16 covers all 4th root of -1 cases.
[/CODE]

In fact,. The general listing is:

-1*-1*-1*1+1 \\ 4 ways if possible for all possibilities
-1*1*1*1+1 \\ 4 ways
sqrt(-1)*sqrt(-1)*-1*-1+1 \\ 6 ways; 12 ways if you use both sqrts possible
sqrt(-1)*sqrt(-1)*1*1+1 \\ 6 ways; 12 ways if ...
sqrt(-1)*sqrt(-1)*sqrt(-1)*(-sqrt(-1))+1 \\ 4 ways; 8 ways if ...
sqrt(sqrt(-1))*sqrt(sqrt(-1))*sqrt(sqrt(-1))*sqrt(sqrt(-1))+1 \\ 1 way; maybe 4 if you use all 4 fourth roots.

And maybe more.


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