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Cubic Equation (Posted on 2013-06-09) Difficulty: 4 of 5
Determine all triplets (a,b,c) of nonzero integers satisfying:
987654321*a + 123456789*b + c = (a + b + c)3

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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re(4): an open letter to Charlie Comment 13 of 13 |
(In reply to re(3): an open letter to Charlie by Steve Herman)

Yes, the below program goes much faster, even in interpreted UBASIC, needed because some intermediate results used in solving for a and b exceed the capacity of QB64.

 20   ' x = a+b+c
 30   for X=0 to 1000000
 40       Rhs=X*X*X-X
 50
 60       if Rhs @ 4 = 0 then
 70           :A=-7716049*Rhs//4:B=61728393*Rhs//4
 80           :Q=int(abs(A)/30864197)
 90           :A=A+Q*30864197:B=B-Q*246913580
100           :while A<0
110           :A=A+30864197:B=B-246913580
120           :wend
130           :while B>=0
140           :C=X-A-B
150           :if C>=0 then print A,B,C
160           :A=A+30864197:B=B-246913580
170           :wend
190   next X

0       0       0
0       0       1
450     4500    5050
7350    6000    6650
27150   1500    1350

About 1 second run time (under 1 second for all the solutins to appear).

The Euclidean algorithm for finding GCD was used to find the constants that multiply by the coefficients to produce the GCD, 4, and zero.


  Posted by Charlie on 2013-06-13 18:01:36
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