Sum of Two Different PrimesDate: 02/22/2002 at 10:36:26 From: Nate Subject: Prime numbers Can the sum of two different primes ever be a factor of the product of those two primes? I.e., Do there exist primes p and q such that (p+q)/pq? Date: 02/22/2002 at 23:00:10 From: Doctor Paul Subject: Re: Prime Numbers There do not exist distinct primes p and q such that (p+q) divides the product p*q. To see why this must be the case, think about the divisors of p*q. They are 1, p, q, and p*q. Clearly p+q is greater than 1, p, and q, so the only way that p+q could be a factor of p*q would be if p+q = p*q. Divide both sides of this equation by p*q to obtain: 1/q + 1/p = 1 If p is the smaller of the two primes, then the smallest p can be is two, and the smallest q can be is three. In this case, 1/q + 1/p = 1/3 + 1/2 = 5/6 < 1 Any other choices of p and q will necessarily be larger, which will make the sum 1/q + 1/p even farther away from one. Thus 1/q + 1/p can never be one if p and q are distinct primes. This completes the proof. I hope this helps. Please write back if you'd like to talk about this more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ |
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