- **Math**
(*https://www.mersenneforum.org/forumdisplay.php?f=8*)

- - **Smallest prime of the form a^2^m + b^2^m, m>=14**
(*https://www.mersenneforum.org/showthread.php?t=23160*)

[QUOTE=JeppeSN;482422]
The next line takes more time. The question is: Hasn't this been considered before?! /JeppeSN[/QUOTE] Are you computing with PFGW :question: Edit: I ran PFGW for bases less than or equal to 50 and found no PRP: [CODE]cat Jeppe.abc2 ABC2 $a^16384+$b^16384 a: from 2 to 50 step 2 b: from 3 to 50 step 2 [/CODE] [CODE]./pfgw64 -f -N Jeppe.abc2[/CODE] |

[QUOTE=JeppeSN;482422]Me:
To be explicit, this is what brute force finds: [CODE]m=0, 2^1 + 1^1 m=1, 2^2 + 1^2 m=2, 2^4 + 1^4 m=3, 2^8 + 1^8 m=4, 2^16 + 1^16 m=5, 9^32 + 8^32 m=6, 11^64 + 8^64 m=7, 27^128 + 20^128 m=8, 14^256 + 5^256 m=9, 13^512 + 2^512 m=10, 47^1024 + 26^1024 m=11, 22^2048 + 3^2048 m=12, 53^4096 + 2^4096 m=13, 72^8192 + 43^8192[/CODE] The next line takes more time. The question is: Hasn't this been considered before?! /JeppeSN[/QUOTE] Probably has, think modular tricks like fermats little theorem and extentions. Mod 8 it becomes a^0+b^0 for example, mod 9 it's a^4+b^4 these work for all bases coprime to 8 or 9 respectively. |

[QUOTE=paulunderwood;482426]Are you computing with PFGW :question:
[/QUOTE] See [URL="https://oeis.org/A291944"]A291944 in OEIS[/URL]; it is not public yet, so see its history. I used PARI/GP [FONT="Fixedsys"]ispseudoprime[/FONT] in a loop, like the code shown there, and I suspect Robert G. Wilson v used Mathematica. Maybe PFGW is faster? There is no point in all of us running the same tests, except whoever uses the best tools will "win" the competition. I just thought maybe this had been established already. /JeppeSN |

[QUOTE=JeppeSN;482473]There is no point in all of us running the same tests, except whoever uses the best tools will "win" the competition. I just thought maybe this had been established already.[/QUOTE]
I will try to write a custom sieve during the weekend. After that I can post the sieve output here, so that interested people can test ranges. PFGW should indeed be faster than Pari or Mathematica. EDIT:- Testing 71^16384+46^16384, PFGW took about 20s, while Pari took 2mins and change. So PFGW is about 6x faster. |

[QUOTE=axn;482479]I will try to write a custom sieve during the weekend. After that I can post the sieve output here, so that interested people can test ranges.
PFGW should indeed be faster than Pari or Mathematica. EDIT:- Testing 71^16384+46^16384, PFGW took about 20s, while Pari took 2mins and change. So PFGW is about 6x faster.[/QUOTE] PFGW will be more than 6x faster with much bigger numbers :smile: |

Something to note:
Use here the convention [$]a > b > 0[/$]. There is a slight chance that the smallest odd prime [$]a^{16384}+b^{16384}[/$] does not minimize [$]a[/$]. As an example, [$]677 < 678[/$], but still [$]677^{128}+670^{128} > 678^{128}+97^{128}[/$] (both of these sums of like powers are prime). However, for the smallest one with that exponent, [$]27^{128}+20^{128}[/$], the value [$]a=27[/$] is also minimal. And I think this will be the case generally, because the bases [$]a[/$] and [$]b[/$] will be relatively small (I conjecture). But we will check for that with 16384 once axn's excellent initiative has come to fruition. /JeppeSN |

[QUOTE=axn;482479]I will try to write a custom sieve during the weekend. After that I can post the sieve output here, so that interested people can test ranges.
PFGW should indeed be faster than Pari or Mathematica. EDIT:- Testing 71^16384+46^16384, PFGW took about 20s, while Pari took 2mins and change. So PFGW is about 6x faster.[/QUOTE] I'm already working on a<b<=1000, or b<a<=1000 which ever convention you use I used fbncsieve to sieve the factors k*2^14+1. It took only ~2min up to k=10^9. Then I used these prime factors in a quickly written GMP program to sieve an array 1000x1000 of a,b. First I removed all values where b>=a, a<2, b<2, a%2=b%2 (both odd or both even), and gcd(a,b)>1. Down to 61K candidates at k=462M. I'm running pfgw while continuing to trial factor. So far no PRP in 2<=b<=16 and b<a<=1000. |

[QUOTE=ATH;482490]I'm already working on a<b<=1000, or b<a<=1000 which ever convention you use
I used fbncsieve to sieve the factors k*2^14+1. It took only ~2min up to k=10^9. Then I used these prime factors in a quickly written GMP program to sieve an array 1000x1000 of a,b. First I removed all values where b>=a, a<2, b<2, a%2=b%2 (both odd or both even), and gcd(a,b)>1. Down to 61K candidates at k=462M.[/quote] Cool. But a custom sieve will be much more efficient. Hopefully that will be useful for >= 15. [QUOTE=ATH;482490]I'm running pfgw while continuing to trial factor. So far no PRP in 2<=b<=16 and b<a<=1000.[/QUOTE] Since the objective is to find the smallest, you should test in a different order. 2<=a<=1000, 1<=b<a |

Stating the obvious for the sake of having it stated
a+b | a^q + b^q for all odd q And a+bi | a^q + b^q for all even q So the result will be definitely not prime over the imaginary field. Corrections are welcome.:smile: |

[QUOTE=axn;482493]Since the objective is to find the smallest, you should test in a different order.[/QUOTE]
I was about to say just that. If the (a,b) space is visualized as this triangle: [CODE] (2,1) (3,2) (4,1) (4,3) (5,2) (5,4) (6,1) (6,3) (6,5) (7,2) (7,4) (7,6) (8,1) (8,3) (8,5) (8,7) . . . . . . [/CODE] it is best to search from the top and down, not from left to right. /JeppeSN |

[QUOTE=axn;482493]But a custom sieve will be much more efficient.[/QUOTE]
That was somewhat presumptuous of me. What is the rate at which your sieve is progressing thru the factor candidates? If you can post your sieve source, I can use that as a starting point. |

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