Problems(3)
If (b^{n}-1)(b-1) is perfect square then 8 divides b
Source: Federation of Bosnia, 1. Grades 2008.
4/23/2008
Let be an even positive integer. Assume that there exist integer such that \frac {b^{n} \minus{} 1}{b \minus{} 1} is perfect square.
Prove that is divisible by 8.
modular arithmetic
Well known p^{4}+q^{4}=r^{4}
Source: Federation of Bosnia, 2. Grades 2008.
4/23/2008
Prove that equation p^{4}\plus{}q^{4}\equal{}r^{4} does not have solution in set of prime numbers.
inequalitiesabstract algebranumber theorygreatest common divisorprime numbers
Simple and easy
Source: Federation of Bosnia, 3. and 4. Grades 2008.
4/23/2008
Find all positive integers and such that \frac{a^{4}\plus{}a^{3}\plus{}1}{a^{2}b^{2}\plus{}ab^{2}\plus{}1} is an integer.
number theory proposednumber theory