%I #17 May 21 2022 14:49:40

%S 24,120,175560

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

%C 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].

%C 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.

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

%H David. W. Boyd and Hershy Kisilevsky, <a href="https://msp.org/pjm/1972/40-1/pjm-v40-n1-p04-s.pdf">The diophantine equation u(u+1)(u+2)(u+3) = v(v + 1)(v + 2)</a>, Pacific J. Math. 40 (1972), 23-32.

%e 24 = 2*3*4 = 1*2*3*4;

%e 120 = 4*5*6 = 2*3*4*5;

%e 175560 = 55*56*57 = 19*20*21*22.

%Y Cf. A121234.

%K nonn,full,fini,bref

%O 1,1

%A _Bernard Schott_, Apr 18 2020