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Choose 4 prime numbers such that adding the first of them...

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Question by Creigh
Submitted on 9/15/2003
Related FAQ: sci.math FAQ: Unsolved Problems
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Choose 4 prime numbers such that adding the first of them equals the fourth, for example: 7+11+13=31. Then the sum of the squares of these numbers is always divisible by 4, or am I mistaken?

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Answer by Tapio
Submitted on 1/8/2004
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You are right, because it's well known that for every prime p > 3, p^2 = 12*n+1, where n is some positive integer. Choose any 4 primes, then the sum of the squares equals to
12*n+1 + 12*m+1 + 12*p+1 + 12*q+1 => 12(n+m+p+q)+4 that is clearly divisible by 4.
This results does not require that the sum of 3 primes equals to the 4th prime.

 

Answer by Tapio
Submitted on 1/8/2004
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Consider the case, where choosen primes are 3 and three other primes (p,q,r)>3,like 3+p+q=r.
The sum of the squares equals to: 9+12*n+1+12*m+1+12*s+1=12*(n+m+r)+12, which is also divisible by 4.
Consider the case, where choosen primes are 3 and 3 and two other primes (p,q)>3, like 3+3+p=q.
The sum of the squares equals to:
9+9+12*n+1+12*m+1=12*(n+m)+20, which is also divisible by 4. Thus your problem is valid for 4 ODD primes.

 

Answer by Tapio
Submitted on 1/8/2004
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There is a counter example:
2+2+3=7, because the sum of the prime squares equals to 4+4+9+49=66, which is not divisible by 4. 66/4=16.5
(The case of primes 2 and (p,q,r)>3 is excluded, because: 2+p+q=r is impossible as r results in even number >2 and therefore r is not prime. even+odd+odd=even). Thus your problem is valid only for ODD primes.

 

Answer by coman_razvan
Submitted on 9/12/2004
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except for 2, all the other prime numbers are odds. therefore we can say that x (odd prime number) = 2k+1. then x^2= 4k^2+4k+1

so if we take ANY 4 prime numbers that fulfill the condition above, the sum of their squares can be written like this
(4k^2+4k+1)+(4m^2+4m+1)+(4n^2+4n+1)+(4p^2+4p+1)=...=4*something
the condition about being prime and the sum of the first three being the fourth is just a decoy.ANY 4 ODD NUMBERS WILL DO THE TRICK


 

Answer by babygurl
Submitted on 11/7/2006
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what number-sis divisible by 66

 

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