|
1.證明x^4+y^4=z^2無正整數解。
2.證明對於任何整數n，n^30-n^14-n^18+n^2可被46410整除。
3.If p is any prime other than 2 or 5, prove that p divides infinitely many
of the integers 9, 99, 999, 9999, ...
4.An integer n is said to be perfect if the sum of the positive factors of
n is 2n. Show that if n is an odd perfect number, it has at least 3
distinct prime factors.
5.設p,q均為自然數而且p/q=1-(1/2)+(1/3)-(1/4)+...-(1/1318)+(1/1319)
證明p可被1979所整除
6.Find one pair of positive integers a and b such that :
(i) ab(a+b) is not divisible by 7;
(ii)(a+b)^7-a^7-b^7 is divisible by 7^7.
Justify your answer
7.Find all positive integers (a, b, n, p) of positive integers such that p
is prime and p^n =a^3+b^3
8.Find all positive integers (x,y,z) such that xy+yz+zx=xyz+2
|