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2011 = Sum of Squares II (Posted on 2011-07-19) Difficulty: 3 of 5
For a positive integer N drawn at random between 3 and 202 inclusively, determine the probability that the base N number 2011 is expressible as the sum of squares of two distinct positive integers in at least one way.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 1 of 1

In 58 out of the 200 cases, or 29%, does the desired result hold:

-----------------------decimal notation-----------
base N        squares                     roots
3             9  49                       3  7
8             9  1024                     3  32
12            1444  2025                  38  45
16            3025  5184                  55  72
19            49  13689                   7  117
20            3025  12996                 55  114
24            12544  15129                112  123
27            1369  38025                 37  195
28            17689  26244                133  162
29            9604  39204                 98  198
33            10404  61504                102  248
36            324  93025                  18  305
44            56169  114244               237  338
48            2209  219024                47  468
49            21904  213444               148  462
52            26244  255025               162  505
56            43264  308025               208  555
59            17689  393129               133  627
64            173889  350464              417  592
68            247009  381924              497  618
75            303601  540225              551  735
76            40804  837225               202  915
83            13689  1129969              117  1063
84            23409  1162084              153  1078
88            170569  1192464             413  1092
91            60025  1447209              245  1203
92            632025  925444              795  962
93            26244  1582564              162  1258
100           112225  1887876             335  1374
107           7225  2442969               85  1563
113           2704  2883204               52  1698
117           6400  3196944               80  1788
120           176400  3279721             420  1811
125           181476  3724900             426  1930
128           262144  3932289             512  1983
129           324  4293184                18  2072
131           950625  3545689             975  1883
133           5184  4700224               72  2168
139           804609  4566769             897  2137
140           335241  5152900             579  2270
145           45796  6051600              214  2460
147           38025  6315169              195  2513
148           223729  6260004             473  2502
152           1440000  5583769            1200  2363
157           2160900  5579044            1470  2362
160           463761  7728400             681  2780
161           2196324  6150400            1482  2480
163           613089  8048569             783  2837
168           769129  8714304             877  2952
176           5071504  5832225            2252  2415
180           1069156  10595025           1034  3255
181           5569600  6290064            2360  2508
184           369664  12089529            608  3477
187           1265625  11812969           1125  3437
188           3790809  9498724            1947  3082
189           3755844  9746884            1938  3122
196           82369  14976900             287  3870
197           7225344  8065600            2688  2840

DEFDBL A-Z
CLS
FOR n = 3 TO 202
  v = 2 * n * n * n + n + 1
  sr1 = 1: sq1 = 1: found = 0
  DO
    r = v - sq1
    sr = INT(SQR(r) + .5)
    IF sr * sr = r THEN
      found = 1
      EXIT DO
    END IF
    sr1 = sr1 + 1
    sq1 = sr1 * sr1
  LOOP UNTIL sq1 > v / 2
  IF found = 1 AND sq1 <> r THEN
   ctHit = ctHit + 1
   PRINT n, sq1; r, sr1; sr
   IF ctHit MOD 40 = 0 THEN
      DO: LOOP UNTIL INKEY$ > "": PRINT
   END IF
  END IF
  ct = ct + 1
NEXT n
PRINT
PRINT ctHit; ct, ctHit / ct

 


  Posted by Charlie on 2011-07-19 18:47:32
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