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 12 ways (Posted on 2011-09-08)
Find the lowest number that can be represented as a sum of 2 squares of distinct integers in 12 different ways.

 No Solution Yet Submitted by Ady TZIDON No Rating

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 computer solution | Comment 2 of 4 |

This confirms Dej Mar's findings. The program reports any N with 12 or more ways of achieving the result, counting differing perfect square addends as 4 ways (per assigning differing or the same signs), but equal perfect squares as just 1 way (with one positive and one negative).

`  N           ways 325           12 425           12 625           12 650           12 725           12 845           12 850           12 925           12 1025          12 1105          16 1300          12 1325          12 1445          12 1450          12 1525          12 1625          16 1690          12 1700          12 1825          12 1850          12 1885          16 2050          12 2125          16 2210          16 2225          12 2405          16 2425          12 2465          16 2500          12 2525          12 2600          12 2650          12 2665          16 2725          12 2825          12 2873          12`

DEFDBL A-Z
CLS
FOR n = 2 TO 9999
ct = 0
FOR a = 0 TO INT(SQR(n) + .5)
rest = n - a * a
IF rest > 0 THEN
b = INT(SQR(rest) + .5)
IF b * b = rest THEN
IF b > a THEN ct = ct + 4
IF b = a THEN ct = ct + 1
IF b < a THEN EXIT FOR
END IF
ELSE
EXIT FOR
END IF
NEXT
IF ct >= 12 THEN
PRINT n, ct
solCt = solCt + 1
IF solCt > 35 THEN END
END IF
NEXT

A variant of the program, requiring that the actual squares be distinct (in other words, using only non-negative integers), finds 160225 as the first one having 12 ways of doing so.

` 160225        12 204425        12 226525        12 292825        12 320450        12 337025        12 348725        12 359125        12 386425        12 403325        12 408850        12 416585        12 453050        12 456025        12 469625        12 491725        12 493025        12 499525        12 505325        12 531505        12 535925        12 544765        12 558025        12 574925        12 585650        12 588965        12 602225        12 612625        12 624325        12 637325        12 640900        12 644725        12 653225        12 674050        12 688025        12 690625        12 `

DEFDBL A-Z
CLS
FOR n = 2 TO 999999
ct = 0
FOR a = 0 TO INT(SQR(n) + .5)
rest = n - a * a
IF rest > 0 THEN
b = INT(SQR(rest) + .5)
IF b * b = rest THEN
IF b > a THEN ct = ct + 1
IF b = a THEN ct = ct + 0
IF b < a THEN EXIT FOR
END IF
ELSE
EXIT FOR
END IF
NEXT
IF ct >= 12 THEN
PRINT n, ct
solCt = solCt + 1
IF solCt > 35 THEN END
END IF
NEXT

The actual ways, using squares of non-negative integers, for forming 160225 are:

` a    b       a^2    b^215  400       225  16000032  399       1024  15920176  393       5776  15444981  392       6561  153664113  384      12769  147456140  375      19600  140625175  360      30625  129600183  356      33489  126736216  337      46656  113569228  329      51984  108241252  311      63504  96721265  300      70225  90000`

 Posted by Charlie on 2011-09-08 11:59:17

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