Let a, b, c be positive integers such that
a, b, c, a+bc, a+cb, b+ca, 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=800b making b+ca=800a 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=6k1, 800a=3(2672k) which works for k=133, making a=797.
Then the sums b+ca=3 and a+b+c=1597 are both primes.
b=3 is impossible as the calculated primes must be distinct.
For b=6k1, a+cb=159912k=3(5334k). This is possible for k=133, but then a+cb=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=7996k
a+bc=12k1
a+cb=159512k
b+ca=3
a+b+c=1597
3 is clearly the least prime among these and the greatest is either 1597 or 12k1 which is greater than 1597 when k>133, but then c<=1.
So 1597 is the greatest and the max difference d=15973=1594.

Posted by xdog
on 20130426 21:16:45 