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For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.

Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of $2$ modulo a Mersenne prime $p=2^k-1$ is only $k$, which is not greater than $\frac pN$ for sufficiently large $p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primesThe critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.

Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of $2$ modulo a Mersenne prime $p=2^k-1$ is only $k$, which is not greater than $\frac pN$ for sufficiently large $p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.

Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of $2$ modulo a Mersenne prime $p=2^k-1$ is only $k$, which is not greater than $\frac pN$ for sufficiently large $p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

math texed.
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edit approved Aug 30, 2015 at 10:23
user76479
edit approved Aug 30, 2015 at 10:23
user76479

For p$p$ prime denote by ord_p(2)$\mathsf{ord}_p(2)$ the multiplicative order of 2$2$ modulo p$p$.

Does there exist N > 0$N > 0$ such that, for ALL primes p$p$, ord_p(2)$\mathsf{ord}_p(2)$ is at least (p-1)/N$\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of 2$2$ modulo a Mersenne prime p=2^k-1$p=2^k-1$ is only k$k$, which is not greater than p/N for$\frac pN$ for sufficiently large p$p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

For p prime denote by ord_p(2) the multiplicative order of 2 modulo p.

Does there exist N > 0 such that, for ALL primes p, ord_p(2) is at least (p-1)/N?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of 2 modulo a Mersenne prime p=2^k-1 is only k, which is not greater than p/N for sufficiently large p.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

For $p$ prime denote by $\mathsf{ord}_p(2)$ the multiplicative order of $2$ modulo $p$.

Does there exist $N > 0$ such that, for ALL primes $p$, $\mathsf{ord}_p(2)$ is at least $\frac{(p-1)}{N}$?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of $2$ modulo a Mersenne prime $p=2^k-1$ is only $k$, which is not greater than $\frac pN$ for sufficiently large $p$.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith

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Steven Galbraith
Steven Galbraith
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Lower bounding the multiplicative order of 2 modulo p

For p prime denote by ord_p(2) the multiplicative order of 2 modulo p.

Does there exist N > 0 such that, for ALL primes p, ord_p(2) is at least (p-1)/N?

If there are infinitely many Mersenne primes then the answer is "no". Since the order of 2 modulo a Mersenne prime p=2^k-1 is only k, which is not greater than p/N for sufficiently large p.

Is there a proof that the answer is "no"?

It is possible that the following question answers this, but it wasn't clear to me: The critical exponent in the multiplicative order of 2 modulo primes

Thanks Steven Galbraith