Izotov 1995 note sierpinski numbers

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theorem_1: Izotov's theorem that k = t^4 is a Sierpinski number for every positive integer t in an explicit system of congruences modulo the primes 2, 3, 5, 17, 257, 65537, 6700417 and 641, with k 2^n + 1 for n = 4m + 2 composite by an algebraic factorization rather than by a covering prime.

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Izotov, Anatoly S., A note on Sierpiński numbers. Fibonacci Quart. 33 (1995), no. 3, 206--207. No notice is printed on the two scanned pages; the journal's issue page that links the PDF carries "Copyright © 2010 The Fibonacci Association. All rights reserved." (https://www.fq.math.ca/33-3.html), every other right reserved.

Sierpinski proved there are infinitely many odd k with k2^n + 1 composite for all n >= 0, using the covering set {3,5,17,257,641,65537,6700417}. Izotov's Theorem 1 shows that if t satisfies an explicit system of congruences modulo 2, 3, 5, 17, 257, 641, 65537 and 6700417, then k = t^4 is a Sierpinski number. The proof splits n into two cases: for n not of the form 4m+2 some prime of the reduced covering set {3,17,257,641,65537,6700417} divides k2^n + 1, while for n = 4m+2 the value equals 4(t2^m)^4 + 1, which factors algebraically as (t^2 2^{2m+1} + t 2^{m+1} + 1)(t^2 2^{2m+1} - t 2^{m+1} + 1) with both factors exceeding 1. He notes k2^{4m+2} + 1 is congruent to 1 mod 5, so Sierpinski's full covering set is not a covering set for these k; he asks whether there are other Sierpinski numbers analogous to Theorem 1 and suggests that the least Sierpinski number k_0 may have no covering set. Problem 1113 asks whether there are Sierpinski numbers with no finite covering set of primes: the paper gives explicit families whose compositeness is partly algebraic, but it shows only that Sierpinski's set is not a covering set for these k, not that they have no finite covering set.

Source: https://www.fq.math.ca/33-3.html.

Read status. Claims checked: Theorem 1 (p. 206), the remark after its proof and the closing question (p. 207) were read clause by clause on the printed pages, and the proof (pp. 206-207) was followed step by step.

Bears on. #1113: Theorem 1 gives infinitely many Sierpinski numbers k = t^4 whose values k 2^n + 1 at n = 4m + 2 are composite by an algebraic factorization rather than by a covering prime, and the remark after it shows that Sierpinski's set {3, 5, 17, 257, 641, 65537, 6700417} does not cover them. The paper does not exhibit a Sierpinski number with no finite covering set, so it does not answer the problem.

Results. Theorem 1 (p. 206, with the remark, question and suggestion on p. 207).

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