MathDB

13

Part of PEN D Problems

Problems(1)

D 13

Source:

5/25/2007
Let Γ\Gamma consist of all polynomials in xx with integer coefficients. For ff and gg in Γ\Gamma and mm a positive integer, let f≡g(modm)f \equiv g \pmod{m} mean that every coefficient of f−gf-g is an integral multiple of mm. Let nn and pp be positive integers with pp prime. Given that f,g,h,rf,g,h,r and ss are in Γ\Gamma with rf+sg≡1(modp)rf+sg\equiv 1 \pmod{p} and fg≡h(modp)fg \equiv h \pmod{p}, prove that there exist FF and GG in Γ\Gamma with F≡f(modp)F \equiv f \pmod{p}, G≡g(modp)G \equiv g \pmod{p}, and FG≡h(modpn)FG \equiv h \pmod{p^n}.
algebrapolynomialmodular arithmeticcalculusintegrationCongruences