LOYOLA COLLEGE (AUTONOMOUS), CHENNAI-600 034.
M.Sc. DEGREE EXAMINATION – mathematics
FourTh SEMESTER – APRIL 2003
MT 4950/ M 1055 number theory
23.04.2003
1.00 – 4.00 Max: 100 Mark
- (a) (i) If (a,m)=1, prove that af(m) º1(modm). Hence deduce Fermats theorem.
(OR)
(ii) of n ³1 , prove that (8)
- (i) State and prove Wilson’s theorem
(ii) Salve the congruence x2 + x +7 º 0 (mod 189)
(OR)
- (iii) Solve x2 + x +7 º 0(mod 73)
(iv) Reduce the congruence 4x2 +2x +1 º0 (mod5) to the form x2 º a (mod p)
hence find the solutions (17)
- a) (i) Let P be an odd prime with (a, p) = 1. Consider he least non-negative
residues module p of the integers a, 2a, 3a, ….
If n denotes the number of these residues that exceed p/2, then
prove that the Legendre symbol
(OR)
(ii) Find the value of the Legendre symbol
(iii) Find the highest power of 7 that divides 1000! (8)
- b) (i) If p is an odd prime and (a,2p) = 1 then prove that the
Legendre symbol
(ii) Define the Jacobi symbol
(OR)
(iii) If f(n) is a multiplicative function and if then prove that
F(n) in multiplicative.
(iv) Define the Moebius function m(n) and prove that inversion formula that
if for every positive integer n
then F(n) . (17)
- (a) (i) Find all the integral solutions of the equation ax + by =c if they
exist, where a, b, c and integers.
(OR)
(ii) Show that there exists at least one positive solution of ax + by = c if
g = (a, b) satisfies the condition g|c and gc >ab. (8)
(b) (i) Prove that all solutions of 3x +5y =1 can be written in the
form x = 2 + 5t, y = -1 -3t.
(ii) Define a primitive solution of x2 + y2 = z2. Prove that the positive
primitive solutions of x2 + y2 = z2 with y even are give by x = r2 = s2,
y = 2rs, z = r2 + s2, where r and s are arbitrary integers of opposite
parity with r > s >0 and (r,s) =1.
(OR)
(ii) Prove that every positive integer is a sum of four squares of integers. (17)
- (a) (i) If P(n) is the partition function, with the usual notation
prove that pm(n) = pm-1(n) + Pn(n-m) if n ³ m>1.
(ii) using the graph of a partition, prove that the usual notation prove that
the number of partitions of n into m summands is the same as the
number of partitions of n having largest sum m and m.
(OR)
(iii) State Euler’s formula and use it prove that Euler’s identify for
any positive integer n. (8)
(b) (i) If n ³ 0 then prove that
(OR)
(ii) of
- (iii) Prove that for o £ x<1, the series (17)