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.
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.
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