Template:Sequence of the Day for September 28

Intended for: September 28, 2011

Timetable

• First draft entered by Alonso del Arte on August 28, 2011 ✓
• Draft reviewed by Charles R Greathouse IV on September 28, 2011 ✓
• Draft approved by Charles R Greathouse IV on September 28, 2011 ✓

Yesterday's SOTD * Tomorrow's SOTD

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A024670: Numbers that are sums of two distinct positive cubes,

n = x 3 + y 3

.

{ 9, 28, 35, 65, 72, 91, 126, 133, 152, 189, 217, 224, 243, 280, 341, 344, ... }

And yes, 1729 is in this sequence, as it exceeds the minimum requirements for inclusion since it is the smallest integer which is the sum of two [distinct] cubes in two different ways. Also, 1729 is the 52nd term of the sequence. (Which means that if you find a term of the sequence once a week...)

As M. F. Hasler observed, this sequence contains no primes because

x 3 + y 3

has the polynomial factorization

(x + y)  (x 2  −  x y + y 2 )

, thus the sum of two [distinct] positive cubes yields a composite number. Fermat’s last theorem implies that this sequence contains no cubes. It trivially yields an infinity of quasi-cubes

n 3 + 1, n   ≥   2,

since 1 is a cube:

{9, 28, 65, 126, 217, 344, ...}

.