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   Author  Topic: sum of the squares of three consecutive odd prime  (Read 636 times)
Christine
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sum of the squares of three consecutive odd prime  
« on: Jan 21st, 2017, 10:54am »
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83 is a prime number and 83 = 3^2 + 5^2 + 7^2
 
Can you prove that there are no other prime numbers which are the sum of the squares of the three consecutive odd primes?
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rloginunix
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Re: sum of the squares of three consecutive odd pr  
« Reply #1 on: Jan 21st, 2017, 5:40pm »
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The triplet (3, 5, 7) is the only set of "three consecutive odd numbers that are all prime" or, equivalently, the only triplet of prime numbers comprising an arithmetic progression with a common difference equal to 2.
 
Proof.
 
Let x be an odd prime > 3.
 
Consider the numbers x, x + 2 and x + 4.
 
Case 1). Let x = 3q + 1 where q = 2, 4, 6, ... Then the term x + 2 = 3q + 3 = 3(q + 1) is composite.
 
Case 2). Let x = 3q + 2 where q = 1, 3, 5, ... Then the term x + 4 = 3q + 6 = 3(q + 2) is composite. What was required to prove.
 
From where it follows that no other set of "three consecutive odd numbers that are all primes" exist.
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Re: sum of the squares of three consecutive odd pr  
« Reply #2 on: Jan 22nd, 2017, 8:09am »
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In the spirit of answering questions that weren't asked:
 
The stipulation that the primes be odd is not necessary - 4+9+25 is even so not a prime.
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Re: sum of the squares of three consecutive odd pr  
« Reply #3 on: Jan 22nd, 2017, 12:01pm »
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I think I'd interpret the problem as finding pn = pk2 + pk+12 + pk+22 where pi is the ith prime and i>1.
I mean, otherwise, why bring squaring and summing to a prime into it?
 
I don't see any immediate reason why there wouldn't be other solutions, but for primes below a billion it's the only solution.
« Last Edit: Jan 22nd, 2017, 12:23pm by towr » IP Logged

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Re: sum of the squares of three consecutive odd pr  
« Reply #4 on: Jan 22nd, 2017, 1:23pm »
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Proof: Any such triplet would only include primes greater than 3, which are well known to be equal to +/- 1 mod 6. That means their squares are all 1 mod 6, making the sum equal to 3 mod 6 and hence divisible by 3, so not prime.
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