Primes'

Number Theory Level 3

There are two primes known to satisfy the following property

$p^2 \mid (2^{p-1}-1)$

Find the sum of those two primes.

The answer is 4604.

1 solution

Brian Charlesworth
Mar 15, 2019

These are known as Wieferich primes , the only known examples being $1093$ and $3511$ . It is speculated that the list of such primes is infinite, but no others less than $5 \times 10^{17}$ have been found.

I believe the first Wieferich prime should be $1093$ , not $1153$ .

Jordan Cahn - 2 years, 2 months ago

Thanks for catching that. I guess my brain thought it would be cool if the two primes were an emirp pair so it made my fingers type $1153$ , (which is a prime, just not a Wieferich prime, sadly). :/

Brian Charlesworth - 2 years, 2 months ago

Sir, a minor typo, I think you meant "an emirp pair".......Also, check out this .....:)

Aaghaz Mahajan - 2 years, 2 months ago

@Aaghaz Mahajan – Ah, thanks for noticing that; my computer must have autocorrected "emirp" to "emir" (as it does not have "emirp" in its vocabulary) and I didn't notice after posting my comment. Your new problem led to me finding that the topic of base-a Wieferich primes has been studied extensively, so thanks for posting it. :)

Brian Charlesworth - 2 years, 2 months ago

@Brian Charlesworth – Welcome Sir :) Also, do check out my new original problem ....!!

Aaghaz Mahajan - 2 years, 2 months ago

Primes'

The answer is 4604.

1 solution

0 pending reports