Computation of <i>Pe</i> for e=1,2,3,4

Computation of Pe for e=1,2,3,4

	e=1	e=2	e=3	e=4	(e=5)
Number of α for which we consider the norm N_e(α)	7	47	1279	589823	73014444031
same as above, reduced by Remark 1 of Appendix^*1	4[computation^*2]	27[computation^*2]	703[computation^*2]	311295[computation^*2]	37580963839
same as above, reduced by all Appendix^*1	4	23	591[computation^*2]	278783	35433545727
Number of different N_e(α) values obtained by computation	4	17	267	202549
m_e = max N_e(α)	5	81	97393	542909003297
Number of prime number p which divides some norm N_e(α)	2	6	118	104443
same as above, with an odd prime number L such that p=2^e+1L+1	0	1	9	4825
\|Pe\| i.e. same as above, with 2 as a primitive root modulo L	0	1	2	2193
Pe	∅	{41}	{593, 977}	{97, 353, 929,…,194600255393, 205620281249}

* For the definition of Pe, α and Ne(), see [Fujima-Ichimura, Section 3].
*1 For these reductions, see [Appendix], [Note] and [Calculation] (PDFs).
*2 Format of the computation outputs is explained [here(pdf)].

Derivation of expressions of the norm

(15) in the paper : f_e(a₀,…,a_2^e-1) = X_e²+Y_e²

e	1	2	3	4
X_e	a₀	a₀²-a₂²+2a₁a₃	homogeneous polynomials of degree 4 with 43 terms	homogeneous polynomials of degree 8 with 30667 terms
Y_e	a₁	a₁²-a₃²-2a₀a₂	homogeneous polynomials of degree 4 with 43 terms	homogeneous polynomials of degree 8 with 30667 terms
Derivation		see Section 2 in the document^*3	see Section 3 in the document^*3	see Section 4 in the document^*3

*3 [Mathematica document (PDF)]

Note on the class number of the p th cyclotomic field, II
Shoichi FUJIMA and Humio ICHIMURA