A283274 Areas of triples of primitive Pythagorean triangles having the same area.

13123110, 2203385574390, 2570042985510, 8943387723270, 826290896699730, 9381843970167926138271390

Offset: 1

Author

Duncan Moore, Mar 04 2017

Keywords

• nonn
• hard
• more

Comments

The generators for the currently known triples are:

a(1): (5,138),(38,77),(55,78): Charles Shedd, 1945

a(2): (55,3422),(61,3306),(266,2035): Randall L Rathbun, 1986

a(3): (143,2622),(869,1610),(1817,2002): Randall L Rathbun, 1986

a(4): (198,3565),(1166,2201),(2035,2438): Randall L Rathbun, 1986

a(5): (731,10434),(1122,9077),(2465,7238): Dan Hoey, May 18 1990

a(6): (352538,2999447),(1931103,2398838),(3063347,3215070): Duncan Moore, Mar 01 2017

The generator (a,b) gives the Pythagorean triangle (b^2+a^2,b^2-a^2,2ab) with area ab(b^2-a^2).

Examples

			The generators of a(1) give the 3 Pythagorean triangles (19069,19019,1380), (7373,4485,5852) and (9109,3059,8580). They have the areas 19019*1380/2 = 4485*5852/2 = 3059*8580/2 = 13123110 = a(1).
		

References

• R. K. Guy, Unsolved Problems in Number Theory, D21.

Links

• Shyam Sunder Gupta, Number Curiosities, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 23, 567-604.

Crossrefs

Cf. A009111.