Finding the smallest prime factor of $sum_a=1^N a^k$

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It is linked to my previous question, but I wanted a ++ clear explanation:
Suppose we have a huge number of that type with a huge $k$.



$sum_a=1^N a^k =1^k+2^k+3^k+...+N^k$



And we want to find the smallest prime factor.
We want to find the smallest $p$ for which
$sum_a=1^N a^k =1^k+2^k+3^k+...+N^kequiv 0 pmod p$



enter image description here



Let's review some facts about $a^kpmod p$: by Fermat's Little Theorem $a^kequiv 1pmod p$ if $(p-1)mid k∧p∤a$. Otherwise $a^k≢1pmod p$ and in the particular case when $p|a⟹a^k equiv 0pmod p$. If the exponent is a multiple of $p-1$, the powers becomes equal to 1.



Okay, now it's here where I get stuck. How do I continue to find the smallest prime factor? Should I count each $a^k pmod p$? I think that I am missing something but I can't nail it.



Thank you very much.






prime-numbers modular-arithmetic exponentiation prime-factorization







share|cite|improve this question























  • Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
    – marshal craft
    Sep 3 at 10:58










  • Here I use it mainly to mean that it DIVIDES
    – alienflow
    Sep 3 at 10:59










  • Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
    – marshal craft
    Sep 3 at 11:03











  • I would start by considering cases, either it is prime or not.
    – marshal craft
    Sep 3 at 11:04










  • Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
    – marshal craft
    Sep 3 at 11:14














up vote
0
down vote

favorite












It is linked to my previous question, but I wanted a ++ clear explanation:
Suppose we have a huge number of that type with a huge $k$.



$sum_a=1^N a^k =1^k+2^k+3^k+...+N^k$



And we want to find the smallest prime factor.
We want to find the smallest $p$ for which
$sum_a=1^N a^k =1^k+2^k+3^k+...+N^kequiv 0 pmod p$



enter image description here



Let's review some facts about $a^kpmod p$: by Fermat's Little Theorem $a^kequiv 1pmod p$ if $(p-1)mid k∧p∤a$. Otherwise $a^k≢1pmod p$ and in the particular case when $p|a⟹a^k equiv 0pmod p$. If the exponent is a multiple of $p-1$, the powers becomes equal to 1.



Okay, now it's here where I get stuck. How do I continue to find the smallest prime factor? Should I count each $a^k pmod p$? I think that I am missing something but I can't nail it.



Thank you very much.






prime-numbers modular-arithmetic exponentiation prime-factorization







share|cite|improve this question























  • Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
    – marshal craft
    Sep 3 at 10:58










  • Here I use it mainly to mean that it DIVIDES
    – alienflow
    Sep 3 at 10:59










  • Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
    – marshal craft
    Sep 3 at 11:03











  • I would start by considering cases, either it is prime or not.
    – marshal craft
    Sep 3 at 11:04










  • Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
    – marshal craft
    Sep 3 at 11:14












up vote
0
down vote

favorite









up vote
0
down vote

favorite











It is linked to my previous question, but I wanted a ++ clear explanation:
Suppose we have a huge number of that type with a huge $k$.



$sum_a=1^N a^k =1^k+2^k+3^k+...+N^k$



And we want to find the smallest prime factor.
We want to find the smallest $p$ for which
$sum_a=1^N a^k =1^k+2^k+3^k+...+N^kequiv 0 pmod p$



enter image description here



Let's review some facts about $a^kpmod p$: by Fermat's Little Theorem $a^kequiv 1pmod p$ if $(p-1)mid k∧p∤a$. Otherwise $a^k≢1pmod p$ and in the particular case when $p|a⟹a^k equiv 0pmod p$. If the exponent is a multiple of $p-1$, the powers becomes equal to 1.



Okay, now it's here where I get stuck. How do I continue to find the smallest prime factor? Should I count each $a^k pmod p$? I think that I am missing something but I can't nail it.



Thank you very much.






prime-numbers modular-arithmetic exponentiation prime-factorization







share|cite|improve this question















It is linked to my previous question, but I wanted a ++ clear explanation:
Suppose we have a huge number of that type with a huge $k$.



$sum_a=1^N a^k =1^k+2^k+3^k+...+N^k$



And we want to find the smallest prime factor.
We want to find the smallest $p$ for which
$sum_a=1^N a^k =1^k+2^k+3^k+...+N^kequiv 0 pmod p$



enter image description here



Let's review some facts about $a^kpmod p$: by Fermat's Little Theorem $a^kequiv 1pmod p$ if $(p-1)mid k∧p∤a$. Otherwise $a^k≢1pmod p$ and in the particular case when $p|a⟹a^k equiv 0pmod p$. If the exponent is a multiple of $p-1$, the powers becomes equal to 1.



Okay, now it's here where I get stuck. How do I continue to find the smallest prime factor? Should I count each $a^k pmod p$? I think that I am missing something but I can't nail it.



Thank you very much.






prime-numbers modular-arithmetic exponentiation prime-factorization




prime-numbers modular-arithmetic exponentiation prime-factorization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




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edited Sep 3 at 10:13









cansomeonehelpmeout

5,5383830




5,5383830










asked Sep 3 at 10:02









alienflow

717




717











  • Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
    – marshal craft
    Sep 3 at 10:58










  • Here I use it mainly to mean that it DIVIDES
    – alienflow
    Sep 3 at 10:59










  • Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
    – marshal craft
    Sep 3 at 11:03











  • I would start by considering cases, either it is prime or not.
    – marshal craft
    Sep 3 at 11:04










  • Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
    – marshal craft
    Sep 3 at 11:14
















  • Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
    – marshal craft
    Sep 3 at 10:58










  • Here I use it mainly to mean that it DIVIDES
    – alienflow
    Sep 3 at 10:59










  • Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
    – marshal craft
    Sep 3 at 11:03











  • I would start by considering cases, either it is prime or not.
    – marshal craft
    Sep 3 at 11:04










  • Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
    – marshal craft
    Sep 3 at 11:14















Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
– marshal craft
Sep 3 at 10:58




Possibly misleading use of the polymorphic $|$ to mean both "such as" and "divides" in cases where either interpretation lead to well formed statements.
– marshal craft
Sep 3 at 10:58












Here I use it mainly to mean that it DIVIDES
– alienflow
Sep 3 at 10:59




Here I use it mainly to mean that it DIVIDES
– alienflow
Sep 3 at 10:59












Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
– marshal craft
Sep 3 at 11:03





Also as i do not know where this question comes from, i assume worst case it is dependent on reiman hypothesis? And you have to divide a whole bunch of times.
– marshal craft
Sep 3 at 11:03













I would start by considering cases, either it is prime or not.
– marshal craft
Sep 3 at 11:04




I would start by considering cases, either it is prime or not.
– marshal craft
Sep 3 at 11:04












Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
– marshal craft
Sep 3 at 11:14




Also maybe related to euler product. Well seems to be it but only for negative $s$ and for finite sums.
– marshal craft
Sep 3 at 11:14















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