A334189 Positive solutions m of the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) = m.

24, 120, 175560

Offset: 1

Author

Bernard Schott, Apr 18 2020

Keywords

• nonn
• full
• fini
• bref

Comments

Boyd and Kisilevsky in 1972 proved that there exist only 3 solutions (x,y) = (2,1), (4,2), (55,19) to the Diophantine equation x * (x+1) * (x+2) = y * (y+1) * (y+2) * (y+3) [see the reference and a proof in the link].

A similar result: in 1963, L. J. Mordell proved that (x,y) = (2,1), (14,5) are the only 2 solutions to the Diophantine equation x * (x+1) = y * (y+1) * (y+2) with 2*3 = 1*2*3 = 6 and 14*15 = 5*6*7 = 210.

Examples

			24 = 2*3*4 = 1*2*3*4;
120 = 4*5*6 = 2*3*4*5;
175560 = 55*56*57 = 19*20*21*22.
		

References

• David Wells, The Penguin Dictionary of Curious and Interesting Numbers (Revised edition), Penguin Books, 1997, entry 175560, p. 175.

Links

• David. W. Boyd and Hershy Kisilevsky, The diophantine equation u(u+1)(u+2)(u+3) = v(v + 1)(v + 2), Pacific J. Math. 40 (1972), 23-32.

Crossrefs

Cf. A121234.