Abstract: Hilbert noted that the polynomial x^4 - 10 x^2 + 1 is irreducible over the integers whereas it is reducible modulo all primes. What is behind this? If two polynomials f,g with integer coefficients take the same set of values modulo all primes, what is the relation between f and g? What proportion of primes divide numbers of the form 2^n + 1? How many of these are of the form 4m+3? What about primes dividing 7^n + 12^n in some arithmetic progression? Are there infinitely many prime numbers such that the decimal expansion of 1/p recurs with period p-1? Given an integer a, if every prime dividing a^n-1 for some n also divides b^n-1, is b necessarily a power of a? We discuss the interesting mathematics behind such questions.
Prof. B. Sury
Indian Statistical Institute, Bangalore
Polynomials, Primes and Progressions