Help Me Prove: (2(p1)(p2)...(pn))^4 + 1 Divisible by Odd Prime q

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Homework Help Overview

The discussion revolves around proving that the expression (2(p1)(p2)...(pn))^4 + 1 is divisible by an odd prime q, where p_i are primes and q is distinct from the p_i's. Participants are exploring the implications of the fundamental theorem of arithmetic in this context.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are attempting to clarify the hypotheses regarding the nature of p_i and the conditions under which the expression is divisible by q. Questions about the divisibility of the expression by the primes p_i and the number 2 are also raised.

Discussion Status

The discussion is ongoing, with some participants providing insights related to the fundamental theorem of arithmetic. There is an exploration of the conditions necessary for the proof, but no consensus has been reached on the specific approach to take.

Contextual Notes

There are assumptions about the nature of the primes involved and the conditions under which the divisibility is to be shown, which may not be fully articulated in the posts.

buzzmath
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I got stuck on one part of this proof. I'm trying to show that
(2(p1)(p2)...(pn))^4 + 1 is divisible by an odd prime q. Can anyone help with some suggestions?
 
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I take it you left out a bunch of hypotheses, such that the p_i are integer and non-zero. The result then follows immediately from the fundamental theorem of arithmetic.
 
p_i are primes and q is an odd prime different from the p_i's
 
The point of Muzza's statement about the "fundamental theorem of arithmetic" is that every number is divisible by a prime number! All you need to do is show that (2(p1)(p2)...(pn))^4 + 1 is not divisible by any of the pi (what would the remainder be?) and that it is not divisible by 2.
 

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