MathDB

49

Part of 1979 IMO Longlists

Problems(1)

Two sequences of integers and condition for prime numbers.

Source: ILL 1979-49

6/5/2011
Let there be given two sequences of integers fi(1),fi(2),⋯(i=1,2)f_i(1), f_i(2), \cdots (i = 1, 2) satisfying: (i)fi(nm)=fi(n)fi(m)(i) f_i(nm) = f_i(n)f_i(m) if gcd⁡(n,m)=1\gcd(n,m) = 1; (ii)(ii) for every prime PP and all k=2,3,4,⋯k = 2, 3, 4, \cdots, fi(Pk)=fi(P)fi(Pk−1)−P2f(Pk−2).f_i(P^k) = f_i(P)f_i(P^{k-1}) - P^2f(P^{k-2}). Moreover, for every prime PP: (iii)f1(P)=2P,(iii) f_1(P) = 2P, (iv)f2(P)<2P.(iv) f_2(P) < 2P. Prove that ∣f2(n)∣<f1(n)|f_2(n)| < f_1(n) for all nn.
inequalitiesnumber theory unsolvednumber theory