Factorial like product











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I am trying to solve http://www.javaist.com/rosecode/problem-519-Factorial-like-Product-askyear-2018



Restating the problem:
$$R(p,k)=prod_{i=1}^{p-1}(i^k+1) hspace{2 mm} text{mod}hspace{2 mm} p$$



$$S(P,k)=sum_{3le ple P}R(p,k)$$



I need to calculate $R(P,k)$ for $P=10^{10}$ and $k=20181026$



For odd $k$, $R(p,k)$ is $0$. For a fixed prime $p$, $R(p,k)$ is periodic with period $p$. And also $R(p,p-1)=1$.



I am unable to proceed further. Any help will be appreciated.










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    You haven't stated the problem. You've merely defined two functions.
    – davidlowryduda
    Nov 23 at 4:25










  • I gave the link to the problem statement and mentioned main points of the problem.
    – Asif
    Nov 23 at 4:27















up vote
1
down vote

favorite
1












I am trying to solve http://www.javaist.com/rosecode/problem-519-Factorial-like-Product-askyear-2018



Restating the problem:
$$R(p,k)=prod_{i=1}^{p-1}(i^k+1) hspace{2 mm} text{mod}hspace{2 mm} p$$



$$S(P,k)=sum_{3le ple P}R(p,k)$$



I need to calculate $R(P,k)$ for $P=10^{10}$ and $k=20181026$



For odd $k$, $R(p,k)$ is $0$. For a fixed prime $p$, $R(p,k)$ is periodic with period $p$. And also $R(p,p-1)=1$.



I am unable to proceed further. Any help will be appreciated.










share|cite|improve this question




















  • 1




    You haven't stated the problem. You've merely defined two functions.
    – davidlowryduda
    Nov 23 at 4:25










  • I gave the link to the problem statement and mentioned main points of the problem.
    – Asif
    Nov 23 at 4:27













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





I am trying to solve http://www.javaist.com/rosecode/problem-519-Factorial-like-Product-askyear-2018



Restating the problem:
$$R(p,k)=prod_{i=1}^{p-1}(i^k+1) hspace{2 mm} text{mod}hspace{2 mm} p$$



$$S(P,k)=sum_{3le ple P}R(p,k)$$



I need to calculate $R(P,k)$ for $P=10^{10}$ and $k=20181026$



For odd $k$, $R(p,k)$ is $0$. For a fixed prime $p$, $R(p,k)$ is periodic with period $p$. And also $R(p,p-1)=1$.



I am unable to proceed further. Any help will be appreciated.










share|cite|improve this question















I am trying to solve http://www.javaist.com/rosecode/problem-519-Factorial-like-Product-askyear-2018



Restating the problem:
$$R(p,k)=prod_{i=1}^{p-1}(i^k+1) hspace{2 mm} text{mod}hspace{2 mm} p$$



$$S(P,k)=sum_{3le ple P}R(p,k)$$



I need to calculate $R(P,k)$ for $P=10^{10}$ and $k=20181026$



For odd $k$, $R(p,k)$ is $0$. For a fixed prime $p$, $R(p,k)$ is periodic with period $p$. And also $R(p,p-1)=1$.



I am unable to proceed further. Any help will be appreciated.







number-theory elementary-number-theory factorial






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edited Nov 23 at 4:28

























asked Nov 23 at 4:19









Asif

1019




1019








  • 1




    You haven't stated the problem. You've merely defined two functions.
    – davidlowryduda
    Nov 23 at 4:25










  • I gave the link to the problem statement and mentioned main points of the problem.
    – Asif
    Nov 23 at 4:27














  • 1




    You haven't stated the problem. You've merely defined two functions.
    – davidlowryduda
    Nov 23 at 4:25










  • I gave the link to the problem statement and mentioned main points of the problem.
    – Asif
    Nov 23 at 4:27








1




1




You haven't stated the problem. You've merely defined two functions.
– davidlowryduda
Nov 23 at 4:25




You haven't stated the problem. You've merely defined two functions.
– davidlowryduda
Nov 23 at 4:25












I gave the link to the problem statement and mentioned main points of the problem.
– Asif
Nov 23 at 4:27




I gave the link to the problem statement and mentioned main points of the problem.
– Asif
Nov 23 at 4:27










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










Notice $k = 2 cdot 2069 cdot 4877 = 2 cdot k'$.



Fix an odd prime $p$, and let $d = (p-1,k')$, so that $k' equiv dx pmod{p-1}$. Since $d$ is the gcd of these two numbers, we have $(x,p-1) = 1$, and thus multiplying by $x$ just permutes the set ${0,...,p-2} bmod{p-1}$, meaning, that for any primitive root $g bmod{p}$,
$$
R(p,k) equiv prod_{i=1}^{p-1}(i^{2k'}+1) equiv prod_{r=0}^{p-2} (g^{2rk'}+1) equiv prod_{r=0}^{p-2}(g^{2drx}+1) equiv prod_{r=0}^{p-2}(g^{2dr}+1) equiv
left( prod_{r=0}^{frac{p-1}{2d}-1} (g^{2dr}+1) right)^{2d} pmod{p}
$$

With the last equivalence since $g^{2dr} = g^{2dr + (p-1)} = g^{2d left( r + frac{p-1}{2d} right)}$.



Note since $p equiv 1 pmod{4}$ iff $-1$ is a quadratic residue $bmod{p}$, we have $R(p,k) = 0$ for all such $p$. Therefore, let $p equiv 3 pmod{4}$, where $-1$ is a nonresidue. Expand the inner product for $R(p,k)$ into elementary symmetric polynomials, $e_n(X)$.



Considering the power sums, we let $X = left{1,g^{2d},g^{4d},...,g^{2dleft(frac{p-1}{2d} - 1right)}right}$, and have, for $n in left{1,...,frac{p-1}{2d}-1 right}$:
$$
p_{n}(X)=sum_{r=0}^{frac{p-1}{2d}-1} g^{2dnr} = frac{g^{2dnleft(frac{p-1}{2d} right)}-1}{g^{2dn}-1}=frac{g^{n(p-1)}-1}{g^{2dn}-1}
$$

As $2d mid p-1$, we have $p ge 2d+1$. For $p>2d+1$, since $2dn le p-(2d+1)$, this implies $p-1 nmid 2dn$, so we have $g^{2dn}-1 notequiv 0 pmod{p}$ while $g^{n(p-1)}-1 equiv 0 pmod{p}$, meaning $p_n(X) equiv 0 pmod{p}$. For $p = 2d+1$, this sum is the empty sum, so is trivially $0$.



Now by Newton's Identities, we have, $ne_n(X) = sum_{i=1}^n (-1)^{i-1}e_{n-i}(X) p_i(X)$, so that $e_n(X) equiv 0 pmod{p}$ for all $n in left{1,...,frac{p-1}{2d} -1 right}$. This means that:



$$
R(p,k) equiv left( left(prod_{r=0}^{frac{p-1}{2d}-1} g^{2dr}right) +1right)^{2d} equiv left(g^{2dleft(1+2+ldots+left(frac{p-1}{2d}-1 right)right)} +1right)^{2d} equiv left(g^{(p-1)left(frac{p-(2d+1)}{4d}right)} + 1 right)^{2d} pmod{p}
$$



Since $p equiv 3 pmod{4}$ and $d$ is odd, we have $p-(1+2d) equiv 3-1-2 equiv 0 pmod{4}$. Therefore $frac{p-(2d+1)}{4d}$ is integral, and thus $R(p,k) equiv 2^{2d} pmod{p}$. Note this same argument works as long as $k'$ is odd, and for any $P$. For odd $k$, as you already noted, the sum is $0$.






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  • Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
    – Asif
    Nov 24 at 6:03












  • Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
    – Asif
    Nov 24 at 6:57








  • 1




    I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
    – Paul LeVan
    Nov 24 at 10:02










  • Thanks a lot :)
    – Asif
    Nov 24 at 10:18











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up vote
1
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Notice $k = 2 cdot 2069 cdot 4877 = 2 cdot k'$.



Fix an odd prime $p$, and let $d = (p-1,k')$, so that $k' equiv dx pmod{p-1}$. Since $d$ is the gcd of these two numbers, we have $(x,p-1) = 1$, and thus multiplying by $x$ just permutes the set ${0,...,p-2} bmod{p-1}$, meaning, that for any primitive root $g bmod{p}$,
$$
R(p,k) equiv prod_{i=1}^{p-1}(i^{2k'}+1) equiv prod_{r=0}^{p-2} (g^{2rk'}+1) equiv prod_{r=0}^{p-2}(g^{2drx}+1) equiv prod_{r=0}^{p-2}(g^{2dr}+1) equiv
left( prod_{r=0}^{frac{p-1}{2d}-1} (g^{2dr}+1) right)^{2d} pmod{p}
$$

With the last equivalence since $g^{2dr} = g^{2dr + (p-1)} = g^{2d left( r + frac{p-1}{2d} right)}$.



Note since $p equiv 1 pmod{4}$ iff $-1$ is a quadratic residue $bmod{p}$, we have $R(p,k) = 0$ for all such $p$. Therefore, let $p equiv 3 pmod{4}$, where $-1$ is a nonresidue. Expand the inner product for $R(p,k)$ into elementary symmetric polynomials, $e_n(X)$.



Considering the power sums, we let $X = left{1,g^{2d},g^{4d},...,g^{2dleft(frac{p-1}{2d} - 1right)}right}$, and have, for $n in left{1,...,frac{p-1}{2d}-1 right}$:
$$
p_{n}(X)=sum_{r=0}^{frac{p-1}{2d}-1} g^{2dnr} = frac{g^{2dnleft(frac{p-1}{2d} right)}-1}{g^{2dn}-1}=frac{g^{n(p-1)}-1}{g^{2dn}-1}
$$

As $2d mid p-1$, we have $p ge 2d+1$. For $p>2d+1$, since $2dn le p-(2d+1)$, this implies $p-1 nmid 2dn$, so we have $g^{2dn}-1 notequiv 0 pmod{p}$ while $g^{n(p-1)}-1 equiv 0 pmod{p}$, meaning $p_n(X) equiv 0 pmod{p}$. For $p = 2d+1$, this sum is the empty sum, so is trivially $0$.



Now by Newton's Identities, we have, $ne_n(X) = sum_{i=1}^n (-1)^{i-1}e_{n-i}(X) p_i(X)$, so that $e_n(X) equiv 0 pmod{p}$ for all $n in left{1,...,frac{p-1}{2d} -1 right}$. This means that:



$$
R(p,k) equiv left( left(prod_{r=0}^{frac{p-1}{2d}-1} g^{2dr}right) +1right)^{2d} equiv left(g^{2dleft(1+2+ldots+left(frac{p-1}{2d}-1 right)right)} +1right)^{2d} equiv left(g^{(p-1)left(frac{p-(2d+1)}{4d}right)} + 1 right)^{2d} pmod{p}
$$



Since $p equiv 3 pmod{4}$ and $d$ is odd, we have $p-(1+2d) equiv 3-1-2 equiv 0 pmod{4}$. Therefore $frac{p-(2d+1)}{4d}$ is integral, and thus $R(p,k) equiv 2^{2d} pmod{p}$. Note this same argument works as long as $k'$ is odd, and for any $P$. For odd $k$, as you already noted, the sum is $0$.






share|cite|improve this answer























  • Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
    – Asif
    Nov 24 at 6:03












  • Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
    – Asif
    Nov 24 at 6:57








  • 1




    I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
    – Paul LeVan
    Nov 24 at 10:02










  • Thanks a lot :)
    – Asif
    Nov 24 at 10:18















up vote
1
down vote



accepted










Notice $k = 2 cdot 2069 cdot 4877 = 2 cdot k'$.



Fix an odd prime $p$, and let $d = (p-1,k')$, so that $k' equiv dx pmod{p-1}$. Since $d$ is the gcd of these two numbers, we have $(x,p-1) = 1$, and thus multiplying by $x$ just permutes the set ${0,...,p-2} bmod{p-1}$, meaning, that for any primitive root $g bmod{p}$,
$$
R(p,k) equiv prod_{i=1}^{p-1}(i^{2k'}+1) equiv prod_{r=0}^{p-2} (g^{2rk'}+1) equiv prod_{r=0}^{p-2}(g^{2drx}+1) equiv prod_{r=0}^{p-2}(g^{2dr}+1) equiv
left( prod_{r=0}^{frac{p-1}{2d}-1} (g^{2dr}+1) right)^{2d} pmod{p}
$$

With the last equivalence since $g^{2dr} = g^{2dr + (p-1)} = g^{2d left( r + frac{p-1}{2d} right)}$.



Note since $p equiv 1 pmod{4}$ iff $-1$ is a quadratic residue $bmod{p}$, we have $R(p,k) = 0$ for all such $p$. Therefore, let $p equiv 3 pmod{4}$, where $-1$ is a nonresidue. Expand the inner product for $R(p,k)$ into elementary symmetric polynomials, $e_n(X)$.



Considering the power sums, we let $X = left{1,g^{2d},g^{4d},...,g^{2dleft(frac{p-1}{2d} - 1right)}right}$, and have, for $n in left{1,...,frac{p-1}{2d}-1 right}$:
$$
p_{n}(X)=sum_{r=0}^{frac{p-1}{2d}-1} g^{2dnr} = frac{g^{2dnleft(frac{p-1}{2d} right)}-1}{g^{2dn}-1}=frac{g^{n(p-1)}-1}{g^{2dn}-1}
$$

As $2d mid p-1$, we have $p ge 2d+1$. For $p>2d+1$, since $2dn le p-(2d+1)$, this implies $p-1 nmid 2dn$, so we have $g^{2dn}-1 notequiv 0 pmod{p}$ while $g^{n(p-1)}-1 equiv 0 pmod{p}$, meaning $p_n(X) equiv 0 pmod{p}$. For $p = 2d+1$, this sum is the empty sum, so is trivially $0$.



Now by Newton's Identities, we have, $ne_n(X) = sum_{i=1}^n (-1)^{i-1}e_{n-i}(X) p_i(X)$, so that $e_n(X) equiv 0 pmod{p}$ for all $n in left{1,...,frac{p-1}{2d} -1 right}$. This means that:



$$
R(p,k) equiv left( left(prod_{r=0}^{frac{p-1}{2d}-1} g^{2dr}right) +1right)^{2d} equiv left(g^{2dleft(1+2+ldots+left(frac{p-1}{2d}-1 right)right)} +1right)^{2d} equiv left(g^{(p-1)left(frac{p-(2d+1)}{4d}right)} + 1 right)^{2d} pmod{p}
$$



Since $p equiv 3 pmod{4}$ and $d$ is odd, we have $p-(1+2d) equiv 3-1-2 equiv 0 pmod{4}$. Therefore $frac{p-(2d+1)}{4d}$ is integral, and thus $R(p,k) equiv 2^{2d} pmod{p}$. Note this same argument works as long as $k'$ is odd, and for any $P$. For odd $k$, as you already noted, the sum is $0$.






share|cite|improve this answer























  • Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
    – Asif
    Nov 24 at 6:03












  • Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
    – Asif
    Nov 24 at 6:57








  • 1




    I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
    – Paul LeVan
    Nov 24 at 10:02










  • Thanks a lot :)
    – Asif
    Nov 24 at 10:18













up vote
1
down vote



accepted







up vote
1
down vote



accepted






Notice $k = 2 cdot 2069 cdot 4877 = 2 cdot k'$.



Fix an odd prime $p$, and let $d = (p-1,k')$, so that $k' equiv dx pmod{p-1}$. Since $d$ is the gcd of these two numbers, we have $(x,p-1) = 1$, and thus multiplying by $x$ just permutes the set ${0,...,p-2} bmod{p-1}$, meaning, that for any primitive root $g bmod{p}$,
$$
R(p,k) equiv prod_{i=1}^{p-1}(i^{2k'}+1) equiv prod_{r=0}^{p-2} (g^{2rk'}+1) equiv prod_{r=0}^{p-2}(g^{2drx}+1) equiv prod_{r=0}^{p-2}(g^{2dr}+1) equiv
left( prod_{r=0}^{frac{p-1}{2d}-1} (g^{2dr}+1) right)^{2d} pmod{p}
$$

With the last equivalence since $g^{2dr} = g^{2dr + (p-1)} = g^{2d left( r + frac{p-1}{2d} right)}$.



Note since $p equiv 1 pmod{4}$ iff $-1$ is a quadratic residue $bmod{p}$, we have $R(p,k) = 0$ for all such $p$. Therefore, let $p equiv 3 pmod{4}$, where $-1$ is a nonresidue. Expand the inner product for $R(p,k)$ into elementary symmetric polynomials, $e_n(X)$.



Considering the power sums, we let $X = left{1,g^{2d},g^{4d},...,g^{2dleft(frac{p-1}{2d} - 1right)}right}$, and have, for $n in left{1,...,frac{p-1}{2d}-1 right}$:
$$
p_{n}(X)=sum_{r=0}^{frac{p-1}{2d}-1} g^{2dnr} = frac{g^{2dnleft(frac{p-1}{2d} right)}-1}{g^{2dn}-1}=frac{g^{n(p-1)}-1}{g^{2dn}-1}
$$

As $2d mid p-1$, we have $p ge 2d+1$. For $p>2d+1$, since $2dn le p-(2d+1)$, this implies $p-1 nmid 2dn$, so we have $g^{2dn}-1 notequiv 0 pmod{p}$ while $g^{n(p-1)}-1 equiv 0 pmod{p}$, meaning $p_n(X) equiv 0 pmod{p}$. For $p = 2d+1$, this sum is the empty sum, so is trivially $0$.



Now by Newton's Identities, we have, $ne_n(X) = sum_{i=1}^n (-1)^{i-1}e_{n-i}(X) p_i(X)$, so that $e_n(X) equiv 0 pmod{p}$ for all $n in left{1,...,frac{p-1}{2d} -1 right}$. This means that:



$$
R(p,k) equiv left( left(prod_{r=0}^{frac{p-1}{2d}-1} g^{2dr}right) +1right)^{2d} equiv left(g^{2dleft(1+2+ldots+left(frac{p-1}{2d}-1 right)right)} +1right)^{2d} equiv left(g^{(p-1)left(frac{p-(2d+1)}{4d}right)} + 1 right)^{2d} pmod{p}
$$



Since $p equiv 3 pmod{4}$ and $d$ is odd, we have $p-(1+2d) equiv 3-1-2 equiv 0 pmod{4}$. Therefore $frac{p-(2d+1)}{4d}$ is integral, and thus $R(p,k) equiv 2^{2d} pmod{p}$. Note this same argument works as long as $k'$ is odd, and for any $P$. For odd $k$, as you already noted, the sum is $0$.






share|cite|improve this answer














Notice $k = 2 cdot 2069 cdot 4877 = 2 cdot k'$.



Fix an odd prime $p$, and let $d = (p-1,k')$, so that $k' equiv dx pmod{p-1}$. Since $d$ is the gcd of these two numbers, we have $(x,p-1) = 1$, and thus multiplying by $x$ just permutes the set ${0,...,p-2} bmod{p-1}$, meaning, that for any primitive root $g bmod{p}$,
$$
R(p,k) equiv prod_{i=1}^{p-1}(i^{2k'}+1) equiv prod_{r=0}^{p-2} (g^{2rk'}+1) equiv prod_{r=0}^{p-2}(g^{2drx}+1) equiv prod_{r=0}^{p-2}(g^{2dr}+1) equiv
left( prod_{r=0}^{frac{p-1}{2d}-1} (g^{2dr}+1) right)^{2d} pmod{p}
$$

With the last equivalence since $g^{2dr} = g^{2dr + (p-1)} = g^{2d left( r + frac{p-1}{2d} right)}$.



Note since $p equiv 1 pmod{4}$ iff $-1$ is a quadratic residue $bmod{p}$, we have $R(p,k) = 0$ for all such $p$. Therefore, let $p equiv 3 pmod{4}$, where $-1$ is a nonresidue. Expand the inner product for $R(p,k)$ into elementary symmetric polynomials, $e_n(X)$.



Considering the power sums, we let $X = left{1,g^{2d},g^{4d},...,g^{2dleft(frac{p-1}{2d} - 1right)}right}$, and have, for $n in left{1,...,frac{p-1}{2d}-1 right}$:
$$
p_{n}(X)=sum_{r=0}^{frac{p-1}{2d}-1} g^{2dnr} = frac{g^{2dnleft(frac{p-1}{2d} right)}-1}{g^{2dn}-1}=frac{g^{n(p-1)}-1}{g^{2dn}-1}
$$

As $2d mid p-1$, we have $p ge 2d+1$. For $p>2d+1$, since $2dn le p-(2d+1)$, this implies $p-1 nmid 2dn$, so we have $g^{2dn}-1 notequiv 0 pmod{p}$ while $g^{n(p-1)}-1 equiv 0 pmod{p}$, meaning $p_n(X) equiv 0 pmod{p}$. For $p = 2d+1$, this sum is the empty sum, so is trivially $0$.



Now by Newton's Identities, we have, $ne_n(X) = sum_{i=1}^n (-1)^{i-1}e_{n-i}(X) p_i(X)$, so that $e_n(X) equiv 0 pmod{p}$ for all $n in left{1,...,frac{p-1}{2d} -1 right}$. This means that:



$$
R(p,k) equiv left( left(prod_{r=0}^{frac{p-1}{2d}-1} g^{2dr}right) +1right)^{2d} equiv left(g^{2dleft(1+2+ldots+left(frac{p-1}{2d}-1 right)right)} +1right)^{2d} equiv left(g^{(p-1)left(frac{p-(2d+1)}{4d}right)} + 1 right)^{2d} pmod{p}
$$



Since $p equiv 3 pmod{4}$ and $d$ is odd, we have $p-(1+2d) equiv 3-1-2 equiv 0 pmod{4}$. Therefore $frac{p-(2d+1)}{4d}$ is integral, and thus $R(p,k) equiv 2^{2d} pmod{p}$. Note this same argument works as long as $k'$ is odd, and for any $P$. For odd $k$, as you already noted, the sum is $0$.







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edited Nov 24 at 18:22

























answered Nov 23 at 11:01









Paul LeVan

8711715




8711715












  • Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
    – Asif
    Nov 24 at 6:03












  • Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
    – Asif
    Nov 24 at 6:57








  • 1




    I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
    – Paul LeVan
    Nov 24 at 10:02










  • Thanks a lot :)
    – Asif
    Nov 24 at 10:18


















  • Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
    – Asif
    Nov 24 at 6:03












  • Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
    – Asif
    Nov 24 at 6:57








  • 1




    I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
    – Paul LeVan
    Nov 24 at 10:02










  • Thanks a lot :)
    – Asif
    Nov 24 at 10:18
















Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
– Asif
Nov 24 at 6:03






Thanks for the detailed answer. There are other primes $p>4139$ which are not coprime to $k'$. Let $m=(k',p-1)$. For those primes is $R(p,k)=2^{2m} hspace{2 mm}text{mod}hspace{2 mm}p$?
– Asif
Nov 24 at 6:03














Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
– Asif
Nov 24 at 6:57






Sorry I meant $R(p,k)=g^{2m}hspace{2 mm}text{mod}hspace{2 mm}p$, where $g$ is a primitive root of $p$.
– Asif
Nov 24 at 6:57






1




1




I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
– Paul LeVan
Nov 24 at 10:02




I was only thinking about $p-1 mid k$ rather than not coprime, but yes, I've updated the argument to show the full result.
– Paul LeVan
Nov 24 at 10:02












Thanks a lot :)
– Asif
Nov 24 at 10:18




Thanks a lot :)
– Asif
Nov 24 at 10:18


















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