So many primes

Number Theory Level 5

We are given three integers a , b , a, b, and c c such that a , b , c , a + b − c , a + c − b , b + c − a , a, b, c, a+b-c, a+c-b, b+c-a, and a + b + c a+b+c are 7 distinct primes. Let d d be the difference between the largest and smallest of these 7 primes. Suppose that 800 ∈ a + b , b + c , c + a . 800∈{a+b, b+c, c+a}. Determine the maximum possible value of d d .

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The answer is 1594.

Sean Ty by Sean Ty , 21, Philippines

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1 solution

Aaaaaa Bbbbbb
Jun 22, 2014

Assume: a>b>c: b+c=800=787+13=797+3 a=797, b=787, c=13, a+b=1584, a+b-c=1571, b+c-a=3, c+a-b=23 are distinct primes. d m a x = a + b + c − ( b + c − a ) = 1594 d_{max}=a+b+c-(b+c-a)=\boxed{1594}

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