Primes p dividing $10^frac{p-1}{3}-1$
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The primes $73$, $12763$ and $255127$, formed by the concatenation of two consecutive Mersenne numbers, have this property:
$73$ divides $10^frac{73-1}{3}-1$
$12763$ divides $10^frac{12763-1}{3}-1$
$255127$ divides $10^frac{255127-1}{3}-1$
I wonder if there are other primes p which divide $10^frac{p-1}{3}-1$ and in particular if there are other primes formed by the concatenation of two consecutive Mersenne numbers with that property.
number-theory
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The primes $73$, $12763$ and $255127$, formed by the concatenation of two consecutive Mersenne numbers, have this property:
$73$ divides $10^frac{73-1}{3}-1$
$12763$ divides $10^frac{12763-1}{3}-1$
$255127$ divides $10^frac{255127-1}{3}-1$
I wonder if there are other primes p which divide $10^frac{p-1}{3}-1$ and in particular if there are other primes formed by the concatenation of two consecutive Mersenne numbers with that property.
number-theory
2
$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
The primes $73$, $12763$ and $255127$, formed by the concatenation of two consecutive Mersenne numbers, have this property:
$73$ divides $10^frac{73-1}{3}-1$
$12763$ divides $10^frac{12763-1}{3}-1$
$255127$ divides $10^frac{255127-1}{3}-1$
I wonder if there are other primes p which divide $10^frac{p-1}{3}-1$ and in particular if there are other primes formed by the concatenation of two consecutive Mersenne numbers with that property.
number-theory
The primes $73$, $12763$ and $255127$, formed by the concatenation of two consecutive Mersenne numbers, have this property:
$73$ divides $10^frac{73-1}{3}-1$
$12763$ divides $10^frac{12763-1}{3}-1$
$255127$ divides $10^frac{255127-1}{3}-1$
I wonder if there are other primes p which divide $10^frac{p-1}{3}-1$ and in particular if there are other primes formed by the concatenation of two consecutive Mersenne numbers with that property.
number-theory
number-theory
asked Nov 26 at 4:51
user613967
2
$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58
add a comment |
2
$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58
2
2
$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58
$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58
add a comment |
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$p$ divides $10^{(p-1)/3}-1$ if and only if $10$ is a cubic residue modulo $p$.
– Jyrki Lahtonen
Nov 26 at 4:58