A099809 Let a,b be prime numbers satisfying the Diophantine equation a^3+b^3=(a+b)*(a^2-a*b+b^2)=c^2. Then the second factor a^2-a*b+b^2 is 3*e^2 for some integer e. This sequence tabulates the 'e' values, sorted by magnitude of c.

19, 4513, 14689, 32401, 26929, 48019, 44641, 72739, 124099, 179683, 211249, 288979, 395089, 386131, 587233, 905059, 1040419, 1410049, 2237011, 1919779, 2078209, 2220451, 2950963, 2767489, 4919971, 5582449, 5019889, 5255761

Offset

0,1

Comments

For each n let a=A099806[n], b=A099807[n], c/12=A098970. Then a^3+b^3=c^2. The left side factors as (a+b)*(a^2-a*b+b^2). The second factor is 3*e^2 for some integer e. The sequence tabluates the 'e' values. These 'e' values all have the form 3*M^4+N^4, for some pair M,N of relatively prime integers of opposite parity. Remember, a and b are prime numbers.

Links

Table of n, a(n) for n=0..27.

James Buddenhagen, Two Primes Cubed which Sum to a Square.

Example

11^3+37^3=228^2, 11^2-11*37+37^2=3*e^2 with e=19, so 19 is in the sequence.

Crossrefs

Cf. A099806, A099807, A098970, A099808.

Sequence in context: A238563 A281820 A213450 * A271588 A233464 A145214

Adjacent sequences: A099806 A099807 A099808 * A099810 A099811 A099812

Keyword

nonn

Author

James R. Buddenhagen, Oct 26 2004

Status

approved