Solution to Problem #3



Al Zimmermann of Lausanne (Switzerland), Sander Post of Ottawa, Ontario (Canada), Philippe Fondanaiche of Paris (France), Carlos Rivera of Nuevo Leon (Mexico), Momo Jeng of Edwardsville IL, Denis Borris of Ottawa, Ontario (Canada), and Chris Welty of London (UK) found solutions. Only one number was found that was the sum of consecutive squares in four ways (554503705). It is apparently unknown whether there exist integers that can be written as a sum of consecutive squares in arbitrarily many ways. If we denote the sum of consecutive squares beginning with n and ending with m by f (n,m), then a summary of the results is:

Number Representation as the Sum of Consecutive Squares
147441 f (85, 101), f (29, 77), f (18, 76)
910805 f (550, 552), f (144, 178), f (35, 140)
1026745 f (716, 717), f (51, 147), f (1, 145)
2403800 f (583, 589), f (368, 384), f (298, 322)
2513434 f (473, 483), f (286, 313), f (66, 198)
3198550 f (225, 275), f (127, 226), f (1, 212)
11739805 f (1079, 1088), f (281, 385), f (196, 349)
15053585 f (2743, 2744), f (933, 949), f (209, 378)
18646301 f (413, 501), f (199, 399), f (29, 382)
33313175 f (433, 565), f (117, 466), f (66, 464)
136448235 f (1548, 1602), f (1293, 1369), f (1, 742)
554503705 f (7442, 7451), f (3613, 3654), f (3570, 3612), f (480, 1210)
962822910 f (988, 1567), f (1071, 1602), f (2398, 2554)
1031517930 f (118, 1457), f (1977, 2211), f (2311, 2489)



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