When do primes and integers in modulo 2^n form equal products?

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SUMMARY

The discussion centers on the conditions under which the equation p1^m1 mod 2^n = p2^m2 mod 2^n holds true for primes p1 and p2 and integers m1 and m2. It concludes that the equality is straightforward and does not inherently relate to the primality of the numbers involved. Additionally, the conversation touches on the abstract algebraic structure of odd numbers under multiplication, noting their isomorphism to a product of groups, specifically a group with 2 elements and a cyclic group with 2n-2 elements.

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John Creighto
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Let p1 and p2 be primes and m1 and m2 be integers when is:

When is p1^m1 mod 2^n = p2^m2 mod 2^n true?

I think this problem has applications to hash-tables.
 
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They're equal when they're equal, there really isn't much to say. I don't see what being prime has to do with it.

As an abstract Abelian group, the odd numbers with multiplication are isomorphic to the product of the group with 2 elements and the cyclic group with 2n-2 elements.
 

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