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Suppose ???? is a √ nonsquare integer, and let ???? denote a fixed square root of ???? from the algebraic closure of ???? . Show that the roots of ???????? (????, ????) ∈ Z[???? ] in the algebraic closure of ???? are precisely the elements √ ???? +1 ???? , ???? −1 where ???? runs through the primitive ????th roots of unity. (c) Suppose ???? is as in (b), and let ???? be a prime for which ????(∤ 2????????. Show ) that ???? is a prime divisor of ???????? (????, ????) if and only if ???? ≡ ???????? (mod ????). (d) Show that if ???? ≡ −1 (mod 4) is a prime divisor of ???????? (????, −1) which does not divide ????, then ???? ≡ −1 (mod ????).

Yet upon inspection we realize we are once again looking at a result that properly belongs not to number theory but to computability theory (or logic); an analogous statement is true if we replace the set of primes with any listable set. Here a set of positive integers ???? is called listable if there is a computer program which, when left running forever, outputs precisely the elements of ????. A very approachable introduction to this circle of ideas is Matijasevich’s article [Mat99]; for complete details see [Mat93].

31. Suppose ???? ≡ 3 (mod 4) is prime. Prove that if 2???? + 1 is also prime, then 2???? + 1 ∣ 2???? − 1. 27. 32. (Selfridge; cf. [Erd50b]) Let ???? ∈ N. Show that 78557⋅2???? + 1 is divisible by some prime number from the set {3, 5, 7, 13, 19, 37, 73} . In particular, 78557 ⋅ 2???? + 1 is always composite. Exercises 43 Table 1. Mann-Shanks criterion: Columns containing only bold entries are indexed by prime numbers. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 1 1 1 1 2 1 2 1 1 3 3 1 3 1 4 6 4 1 4 5 1 5 10 10 1 6 6 33.

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Not Always Buried Deep by Paul Pollack


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