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,

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.

Contributions came from Flavio Torasso, Emmanuel Vantieghem


Flavio wrote:

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

Q1:   A=65536
Q2:   a=16


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.



Records   |  Conjectures  |  Problems  |  Puzzles