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Some special sums (Posted on 2015-02-05) Difficulty: 2 of 5
Show that all sums of two consecutive odd prime numbers have at least three prime factors, not necessarily distinct. Example: 3+5=8. 8's factors: 2,2,2.

No Solution Yet Submitted by Ady TZIDON    
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Solution Odd premise (spoiler) | Comment 1 of 4
The question involves a little bit of attempted misdirection, as this can be proven for any pair of consecutive odd numbers, not just primes.

If the even number that separates them is 2n, then their sum is 4n, so the sum can be factored as 2*2*n, and n must have at least 1 prime factor.  The 2 two's get us up to at least 3 prime factors.

q.e.d

Edited on February 5, 2015, 2:29 pm
  Posted by Steve Herman on 2015-02-05 14:20:16

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