By Leo Moser

ISBN-10: 1931705011

ISBN-13: 9781931705011

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We show that f (pα−1 ) ≡ 0 and f (0) ≡ 0 imply f (kpα−1 ) ≡ 0, k = 2, 3, . . Proof. Since f (0) ≡ 0, we have c0 ≡ 0, f (x) ≡ c1 + c2 x2 + · · · + cm xm , and f (pα−1 ) ≡ c1 pα−1 ≡ 0 (mod pα ), so that c1 ≡ 0 (mod p). But now f (kpα−1 ) ≡ 0, as required. On Relatively Prime Sequences Formed by Iterating Polynomials (Lambek and Moser) Bellman has recently posed the following problem. If p(x) is an irreducible polynomial with integer coefficients and p(x) > x for x > c, prove that {pn (c)} cannot be prime for all large n.

1 = 712 . About 1895 Ramanujan made the same conjecture but no progress towards a solution of the problem. ) problem in the Monthly. No solutions were offered. However in 1950 an incorrect solution was published and since that time several faulty attempts to prove the result have been made. Again, about 1950 someone took the trouble to check, by brute force, the conjecture up to n = 50. However, earlier, in his book on the theory of numbers Kraitchik already had proved the result up to 5000. As far as we know that is where the problems stands.

Landau was quite anxious to have an elementary proof. Though somewhat related results have been given by Whitman and by Carlitz, Dirichlet’s result is quite isolated. Thus, no similar nontrivial result is known for other ranges. In 1896 Aladow, in 1898 von Sterneck, and in 1906 Jacobsthall, took up the question of how many times the combinations ++, +−, −+, and −− appear. They showed that each of the four possibilities appeared, as one might expect, with frequency 1/4. In 1951 Perron examined the question again and proved that similar results hold if, instead of consecutive integers, one considers integers separated by a distance d.

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An Introduction to the Theory of Numbers by Leo Moser


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