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The Seven Primes (Posted on 2013-04-26) Difficulty: 3 of 5
Let a, b, c be positive integers such that a, b, c, a+b-c, a+c-b, b+c-a, a+b+c are 7 distinct primes.

The sum of two of a, b, c is 800.

If d be the difference of the largest prime and the least prime among those 7 primes, find the maximum value of d.

No Solution Yet Submitted by Danish Ahmed Khan    
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analytical solution Comment 2 of 2 |
All the requested primes are distinct so if one of a,b,c =2, the other two are odd.  But then a+b+c is even. Thus each of a,b,c is odd. 

Say b+c=800.  Then c=800-b making b+c-a=800-a and a+b+c=800+a.  

If a=3, a+b+c=803 which is not a prime.

If a=6k+1, 800+a=801+6k=3(267+2k) which is impossible for positive k and prime 800+a. 

Setting a=6k-1, 800-a=3(267-2k) which works for k=133, making a=797.  

Then the sums b+c-a=3 and a+b+c=1597 are both primes.

b=3 is impossible as the calculated primes must be distinct.

For b=6k-1, a+c-b=1599-12k=3(533-4k). This is possible for k=133, but then a+c-b=3 which is already assigned.

So b=6k+1 and values can be assigned to a,b,c and the requested sums.

a=797
b=6k+1
c=799-6k
a+b-c=12k-1
a+c-b=1595-12k
b+c-a=3
a+b+c=1597

3 is clearly the least prime among these and the greatest is either 1597 or 12k-1 which is greater than 1597 when k>133, but then c<=1.

So 1597 is the greatest and the max difference d=1597-3=1594.


  Posted by xdog on 2013-04-26 21:16:45
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