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 Another 2006 Problem (Posted on 2006-09-30)
Determine the total number of positive integer solutions of:

2/x + 3/y = 1/2006

 See The Solution Submitted by K Sengupta No Rating

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 re: A Computer Assisted look ??? | Comment 7 of 8 |
(In reply to A Computer Assisted look ??? by brianjn)

This is a precision problem. Instead of float you should have used double. In the first example you give, x=99068, y=6272, the LHS = 621354488 and the RHS does equal the 621354496 you show, but these are close enough to count as equal for single precision floating point. (LONG for long integers would also probably work, as you are working with integers; and if you exceed the limits of that type, the program would bomb with an overflow rather than give misleading results.)

The following program uses xdog's factorization to list the 72 solutions:

DEFDBL A-Z
OPEN "anot2006.txt" FOR OUTPUT AS #2
FOR p2 = 0 TO 3
FOR p3 = 0 TO 1
FOR p17 = 0 TO 2
FOR p59 = 0 TO 2
f1 = INT(2 ^ p2 * 3 ^ p3 * 17 ^ p17 * 59 ^ p59 + .5)
f2 = 24144216 / f1
IF f2 <> INT(f2) THEN
REM
END IF
x = f1 + 2 * 2006
y = f2 + 3 * 2006
IF x > 0 AND y > 0 THEN
GOSUB report
END IF
x = 2 * 2006 - f1
y = 3 * 2006 - f2
IF x > 0 AND y > 0 THEN
GOSUB report
END IF
NEXT
NEXT
NEXT
NEXT

PRINT ct, big
END

report:
PRINT #2, USING "#########"; x; y
PRINT USING "#########"; x; y
ct = ct + 1
IF x > big THEN big = x
IF y > big THEN big = y
RETURN

Note that it also checks to see if there are solutions that have two negative factors that lead to positive x and y; it didn't find any.

The output file, sorted externally to the program, shows the 72 solutions for x and y:

`     4013 24150234     4014 12078126     4015  8054090     4016  6042072     4018  4030054     4020  3024045     4024  2018036     4029  1426266     4036  1012027     4046   716142     4063   479434     4071   415242     4080   361080     4114   242726     4130   210630     4148   183549     4189   142426     4216   124372     4248   108324     4301    89562     4366    74222     4420    65195     4484    57171     4590    47790     4720    40120     4879    33866     5015    30090     5168    26904     5428    23069     5746    19942     6018    18054     6324    16461     7021    14042     7480    12980     7493    12954     8024    12036    10030    10030    10948     9499    10974     9486    12036     9027    14455     8330    16048     8024    17936     7752    21063     7434    24898     7174    28084     7021    31860     6885    38114     6726    45784     6596    55165     6490    63189     6426    72216     6372    87556     6307   106318     6254   122366     6222   140420     6195   181543     6154   208624     6136   240720     6120   359074     6086   413236     6077   477428     6069   714136     6052  1010021     6042  1424260     6035  2016030     6030  3022039     6026  4028048     6024  6040066     6022  8052084     6021 12076120     6020 24148228     6019`

 Posted by Charlie on 2006-10-01 10:17:41

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