Primes again!

Number Theory Level 3

There are only a finite number of primes known to satisfy the following property

p 2 p^2 divides ( 3 p − 1 − 1 (3^{p-1}-1 )

Find the product of those primes.

Inspiration


The answer is 11066033.

Aaghaz Mahajan by Aaghaz Mahajan , 19, India

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1 solution

Aaghaz Mahajan
Mar 17, 2019

These are known as Mirimanoff primes , the only known examples being 11 11 and 1006003 1006003 .

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What about numbers other than 2&3?? Is there any general form? And p 3 p^3 instead of p 2 p^2 ?

Mr. India - 2 years, 2 months ago

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Here is a list of base-a Wieferich primes. It is conjectured that there an infinite number of primes for each base a, and there appears to be no general form. The issue of p 3 p^{3} and higher orders is discussed later in the linked article.

Brian Charlesworth - 2 years, 2 months ago

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Thank you sir

Mr. India - 2 years, 2 months ago

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