Problems & Puzzles: Puzzles

Puzzle 775. Triangles and consecutive primes Vic Bold sent the following nice puzzle:   Let p1, p2, p3, p4, p5 & p6 to be six consecutive primes in the same order.   Let draw a triangle having the following vertexes: (p1, p2), (p3, p4) & (p5, p6).   Let A to be the surface of this triangle.   We are now looking for primes sextets such that:   a) A is a square (example for p1=2520552823, A=36864, sqrt(A)=192) b) A=2^a (example for p1=4085747551, a=15)   Q1:  Report your prime sextet such that A is your largest square.   Q2: Report your prime sextet such that A is your largest a.

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Contributions came from Flavio Torasso, Emmanuel Vantieghem

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Flavio wrote:

My best prime sextet starts from p1=7177165505351 and answers both questions simultaneously:

Q1:   A=65536

Q2:   a=16

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Emmanuel wrote:

 Q1, I examined all sextets with  p1 < 25*10^9  and this was the one with biggest square A :

{10726904603, 10726904633, 10726904659, 10726905041, 10726905097, 10726905103}

with  A = 296^2.
 

Without systematic search, I found (by hazard ?) :

{1425172824978589009, 1425172824978589031, 1425172824978589069, 1425172824978589487, 1425172824978589597, 1425172824978589601}

with  A = 342^2.
 

But, I think there are smaller sextets with bigger square surface. I could not find a solution with surface a power of 2, greater than 2^15.

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