m0f(m)xm=(1x)d+1g0+g1x+g2x2++gdxd and there is unique rational numbes from g0,,gd and let f(t)=k=0dfktk degree d polynomial...
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m0f(m)xm=(1x)d+1g0+g1x+g2x2++gdxd and there is unique rational numbes from g0,,gd and let f(t)=k=0dfktk degree d polynomial with rational coeffcients Given f(x) is an integer for x=0,,d. (i.e: f(m)=2m(m1) has this property) Show that implies the gk are all integers and when x is an integer then f(x) is an integer. Hint: prove that f(t)=k=0dgk(d+tkd), identity of polynomial in t so we consider system of equation from t=0,,d m0f(m)xm=(1x)d+1g0+g1x+g2x2++gdxd and there is unique rational numbes from g0,,gd and let f(t)=k=0dfktk degree d polynomial with rational coeffcients Given f(x) is an integer for x=0,,d. (i.e: f(m)=2m(m1) has this property) Show that implies the gk are all integers and when x is an integer then f(x) is an integer. Hint: prove that f(t)=k=0dgk(d+tkd), identity of polynomial in t so we consider system of equation from t=0,,d
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