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

Author

James R. Buddenhagen, Oct 26 2004

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A099806, A099807, A098970, A099808.