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

 

 

1. (a) (i)   If  (a,m)=1, prove that a^f(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)

 

2. 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)

 

1. 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)

 

 

 

 

 

3. (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)

 

4. (a) (i) If P(n) is the partition function, with the usual notation

prove that p_m⁽ⁿ⁾ = p_m-1⁽ⁿ⁾ + P_n^(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)

 

 

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