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A108287
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Consider the Diophantine equations x+y=a, x^2+y^2=b, x^3+y^3=c. There are three values of a for each pair b,c with b>0 and c>0. Sequence gives values of b.
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2
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1, 4, 7, 9, 13, 16, 19, 21, 25, 28, 31, 36, 37, 39, 43, 49, 49, 52, 57, 61, 63, 64, 67, 73, 76, 79, 81, 84, 91, 91, 93, 97, 100, 103, 109, 111, 112, 117, 121, 124, 127, 129, 133, 133, 139, 144, 147, 148, 151, 156, 157, 163, 169, 169, 171, 172, 175, 181, 183
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refs;
listen;
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text;
internal format)
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OFFSET
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1,2
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COMMENTS
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The following information was extracted from the Derive script and notes.
The values of x and y are not required to be integers or even real numbers.
Eliminating x and y from the equations gives:
a^3 - 3*a*b + 2*c = 0.
This is the Diophantine equation that is being solved. For a given b and c there will be three possibly complex roots. It is required to find those values of b and c where all three roots are integers.
The values for b are the integers of the form (d^2 + d*e + e^2)/3 where d and e are positive integers and d <= e. This sequence lists these values in order with repetition.
The corresponding values for c are d*e*(d+e)/2 given in A108940, and the three values for a are d, e and -(d+e).
(End)
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LINKS
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PROG
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(Derive) See Links section
(PARI) upto(blim)={my(L=List()); for(e=1, sqrtint(3*blim), for(d=1, e, my(b=(d^2+d*e+e^2)/3); if(b<=blim && !frac(b), listput(L, b)))); listsort(L); Vec(L)} \\ Andrew Howroyd, Mar 13 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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James R FitzSimons (cherry(AT)getnet.net), Jun 22 2007
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EXTENSIONS
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STATUS
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approved
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