MathDB

1

Part of 2003 Federal Competition For Advanced Students, Part 2

Problems(2)

P(n) = n^3 -n^2 -5n+ 2 is a prime

Source: Austrian Mathematical Olympiad 2003, Part 2, D1, P1

6/18/2011
Consider the polynomial P(n)=n3−n2−5n+2P(n) = n^3 -n^2 -5n+ 2. Determine all integers nn for which P(n)2P(n)^2 is a square of a prime.
[hide="Remark."]I'm not sure if the statement of this problem is correct, because if P(n)2P(n)^2 be a square of a prime, then P(n)P(n) should be that prime, and I don't think the problem means that.
algebrapolynomialnumber theory proposednumber theory
Number abc in base g will be cba in base h = g ± 1

Source: Austrian Mathematical Olympiad 2003, Part 2, D2, P1

6/18/2011
Prove that, for any integer g>2g > 2, there is a unique three-digit number abc‾g\overline{abc}_g in base gg whose representation in some base h=g±1h = g \pm 1 is cba‾h\overline{cba}_h.
number theory proposednumber theory