A216682 Perfect squares which can be written in all the four forms a^2+b^2, a^2+2*b^2, a^2+3*b^2 and a^2+7*b^2, with a > 0 and b > 0.

3600, 4624, 12100, 12321, 14400, 18496, 20449, 24336, 26896, 30276, 32400, 37249, 41616, 46225, 48400, 49284, 51076, 57600, 73984, 75076, 81796, 85264, 90000, 97344, 101124, 106929, 107584, 108900, 110889, 112225, 113569, 115600, 121104, 126736, 129600, 139876, 144400, 148225, 148996, 150544, 165649, 166464, 176400, 184041, 184900, 193600, 197136

Offset: 1

Author

V. Raman, Sep 13 2012

Keywords

• nonn

Comments

If a composite number C, say, can be written in the form C = a^2+k*b^2, for some integers a & b, then every prime factor P (for C) being raised to an odd power can be written in the form P = c^2+k*d^2, for some integers c & d.

This statement is only true for k = 1, 2, 3.

For k = 7, with the exception of the prime factor 2, the statement mentioned above is true.

Crossrefs

Cf. A216451, A216500.