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12 ways (Posted on 2011-09-08) Difficulty: 4 of 5
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    
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Solution 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^2
15  400       225  160000
32  399       1024  159201
76  393       5776  154449
81  392       6561  153664
113  384      12769  147456
140  375      19600  140625
175  360      30625  129600
183  356      33489  126736
216  337      46656  113569
228  329      51984  108241
252  311      63504  96721
265  300      70225  90000

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