2007 Mathematical Olympiad Summer Program Tests
... Prove that for all positive integers x and y, the numerator of fp (x) − fp (y), when written in lowest terms, is divisible by p3 . 3. Let n be an integer greater than 2, and P1 , P2 , · · · , Pn distinct points in the plane. Let S denote the union of the segments P1 P2 , P2 P3 , . . . , Pn−1 Pn . De ...
... Prove that for all positive integers x and y, the numerator of fp (x) − fp (y), when written in lowest terms, is divisible by p3 . 3. Let n be an integer greater than 2, and P1 , P2 , · · · , Pn distinct points in the plane. Let S denote the union of the segments P1 P2 , P2 P3 , . . . , Pn−1 Pn . De ...
2. Ideals in Quadratic Number Fields
... The most important examples are abelian groups G: they are all Zmodules via ng = g + . . . + g (n terms) for n > 0 and ng = −(−n)g for n < 0. In particular, a subring M of a commutative ring R is a Z-module; it is alsoan R-module if and only if M is an ideal. If M and N are R-modules, then so is M ⊕ ...
... The most important examples are abelian groups G: they are all Zmodules via ng = g + . . . + g (n terms) for n > 0 and ng = −(−n)g for n < 0. In particular, a subring M of a commutative ring R is a Z-module; it is alsoan R-module if and only if M is an ideal. If M and N are R-modules, then so is M ⊕ ...
