Can the order of 2 modulo $p$ be arbitrarily small (relative to $p-1$) if p is a Wieferich prime
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Let $k={rm ord}_p 2$ be the multiplicative order of 2 modulo p. Can the ratio $frac{p-1}{k}$ be arbitrarily large if $p$ is a Wieferich prime? This is known to be true without the Wieferich restriction (related post) using Chebotarev's density theorem, but what happens if you introduce the restriction that $p$ is a Wieferich prime? Of greater interest to me is to know whether $p=O(k^t)$ for some fixed positive integer $t$ or not.
group-theory number-theory
edited Nov 22 at 18:48
Jyrki Lahtonen
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asked Nov 22 at 17:26
EGME
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Let $k={rm ord}_p 2$ be the multiplicative order of 2 modulo p. Can the ratio $frac{p-1}{k}$ be arbitrarily large if $p$ is a Wieferich prime? This is known to be true without the Wieferich restriction (related post) using Chebotarev's density theorem, but what happens if you introduce the restriction that $p$ is a Wieferich prime? Of greater interest to me is to know whether $p=O(k^t)$ for some fixed positive integer $t$ or not.
group-theory number-theory
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
asked Nov 22 at 17:26
EGME
112
3
Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $k={rm ord}_p 2$ be the multiplicative order of 2 modulo p. Can the ratio $frac{p-1}{k}$ be arbitrarily large if $p$ is a Wieferich prime? This is known to be true without the Wieferich restriction (related post) using Chebotarev's density theorem, but what happens if you introduce the restriction that $p$ is a Wieferich prime? Of greater interest to me is to know whether $p=O(k^t)$ for some fixed positive integer $t$ or not.
group-theory number-theory
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
asked Nov 22 at 17:26
EGME
112
Let $k={rm ord}_p 2$ be the multiplicative order of 2 modulo p. Can the ratio $frac{p-1}{k}$ be arbitrarily large if $p$ is a Wieferich prime? This is known to be true without the Wieferich restriction (related post) using Chebotarev's density theorem, but what happens if you introduce the restriction that $p$ is a Wieferich prime? Of greater interest to me is to know whether $p=O(k^t)$ for some fixed positive integer $t$ or not.
group-theory number-theory
group-theory number-theory
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
asked Nov 22 at 17:26
EGME
112
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
asked Nov 22 at 17:26
EGME
112
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
edited Nov 22 at 18:48
Jyrki Lahtonen
107k12166364
107k12166364
asked Nov 22 at 17:26
EGME
112
asked Nov 22 at 17:26
EGME
112
asked Nov 22 at 17:26
EGME
112
112
3
Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08
add a comment |
3
Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08
3
3
Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08
Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08
add a comment |
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Given that we only know two Wieferich primes, the answer is likely to be unknown.
– Wojowu
Nov 22 at 19:08