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It is a well known fact that almost all polynomials with integer
coefficients are irreducible over rationals. Thus one can naturally expect
to find an irreducible polynomial 'close' to any given polynomial in the
sense that, there is a universal constant 'C' such that given a polynomial
f(x) with integer coefficients, there is an irreducible polynomial g(x)
with integer coefficients and degree at most that of f(x), such that the
'distance' (the notion of distance between polynomials to be described
during the talk) between f(x) and g(x) is bounded by 'C'. This was
proposed by Paul Turan during the 70's. Though the problem remains open,
Andrzej Schinzel (in 1970), Michael Filaseta and myself have been able to
provide partial answers to the problem. I shall discuss our result as well
as a connection to another open problem from elementary number theory.
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