* J. London Math. Soc. 15 (1940), 159–60.Google Scholar

† It is not supposed that k ≥ 3; indeed k may be equal to zero.

* Suppose on the contrary that α is a rational multiple of π. Let p be such that pα is an integral multiple of 2π. Now the polynomial has integer coefficients, and has two equal roots, on account of the choice of p. Hence is an algebraic integer of degree 1 or 2. But it is not of degree 1 since and it is not of degree 2 either, since the constant term in is equal to which is not a rational number. Hence α is an irrational multiple of π.

* A necessary and sufficient condition to ensure the irreducibility of P(x) is that

{a ² + (b + m ² + 2)²} (b + 2 + 2a) (b + 2 − 2a) (a ² − 4b + 8 − m ²) ≠ 0

for m = 0,1,2,3, …; a sufficient condition to secure that the equation P(x) = 0 has two real positive roots and two non-real roots is that 0 > b ≥ a.

† If α were a rational multiple of π, then P(x) would be divisible by a cyclotomic polynomial, which is not possible since P(x) is irreducible and has zeros which do not lie on the unit circle.

* Acta Math. 30 (1906), 369.Google Scholar

† Math. Annalen 77 (1916), 510–12Google Scholar. See also Pólya, G. and Szegö, G., Aufgaben und Lehrsätze aus der Analysis, 2 (1925), 142 and 357.Google Scholar

‡ Hardy, G. H., J. Indian Math. Soc. 11 (1919), 162–166Google Scholar, gives the special case of this theorem in which the sequence is assumed to have one limit point only.

* As soon as it was proved that b ₁b ₂ … b _l ≠ 0 we could have deduced from the theorem mentioned below that ξ₁, ξ₂, …, ξ₁ are algebraic integers. Let g ₁(z), g ₂(z), …, g _m(z) be non-vanishing integral functions of z, and let M _ν(z) denote the maximum modulus of g _ν(z) on the circle | z | = r. If r ⁻¹ log M _ν(r) → 0 as r → ∞ (ν = 1, 2, …, m), and

has rational integral values for z = 0, 1, 2, 3, …, then g ₁(z), g ₂(z), …, g _m(z) are polynomials and ξ₁, ξ₂, …, ξ_mare algebraic integers [Pólya, G., Math. Annalen, 77 (1916), 512CrossRefGoogle Scholar]. In connexion with this theorem, it may be stated that it is also true that, if a number η is a conjugate of one or more of the numbers ξ₁, ξ₂, …, ξ_m, then η is one of the numbers ξ₁, ξ₂, …, ξ_m.

* Since

* Pólya, G. and Szegö, G., Aufgaben, 2 (1925), 149 and 368.Google Scholar