PMA305
SCHOOL OF MATHEMATICS AND STATISTICS
Topics in Number Theory
Autumn Semester
2008-2009
2 hours 30 minutes
Answer four questions. If you answer more than four questions, only your best four will
be counted.
No credit will be given for solutions which rely solely on the use of a calculator. Your
solutions should give enough details to make it clear how you arrived at the answer.
1
(i)
State Fermat's Little Theorem and show that
n1729 ≡ n (mod 1729)
(9 marks)
for all integers n.
(ii)
Decode it.
In the RSA cryptosystem, you publish (n, e) = (187, 7) and receive 13.
(10 marks)
(iii) Determine the remainder when 27! is divided by 31. [No credit will be
given for a solution which does not use Wilson's Theorem.]
(6 marks)
2
(i)
State The Law of Quadratic Reciprocity and use it to determine whether
the congruence
x2 + 3x + 31 ≡ 0 (mod 103)
(16 marks)
has a solution. If it has, solve it.
(ii)
Let p be a prime number greater than 2. Show that the congruence
x4 + 3x2 + 2 ≡ 0 (mod p)
µ ¶
µ ¶
−1
−2
has a solution if and only if either
= 1 or
= 1. Deduce that it has no
p
p
solution if and only if p ≡ 7 (mod 8).
(9 marks)
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Turn Over
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(i)
by 641.
(ii)
For each of the numbers given, nd a prime number which divides it:
(a)
2r − 1 where r is an even natural number,
(b)
2s + 1 where s is an odd natural number,
(c)
2121 − 1
(d)
2124 + 1.
(9 marks)
Factorize 6404 into its prime factors and hence show that 232 + 1 is divisible
(5 marks)
(iii) State a formula which gives all even perfect numbers, and show that every
even perfect number is the sum of the rst q natural numbers for some prime number q .
(You may quote the formula
1 + 2 + . . . + n = 12 n(n + 1)
for the sum of the rst n natural numbers.)
(iv)
Show that
(a)
(5 marks)
Let m and n be integers such that 1 < m < n and m divides n.
σ(n)
1
>1+ .
n
m
(b) If n is a natural number and p is a prime number such that pσ(n) =
nσ(p), show that p divides n and use part (a) to deduce that n = p. [σ(n) denotes the
sum of the positive divisors of n.]
(6 marks)
4
Let x, y, z be a primitive Pythagorean triple. Show that one of x, y is even and the
other is odd, and state formulae which describe all primitive Pythagorean triples with x
even.
(5 marks)
(a) Find all primitive Pythagorean triples which include 333.
(10 marks)
(b) Find two non-primitive Pythagorean triples which include 333.
(2 marks)
(c) Is there a primitive Pythagorean triple which includes 338?
(2 marks)
(d) Is there a non-primitive Pythagorean triple which includes 338? (2 marks)
(e) Is there a Pythagorean triple in which two of the numbers are squares? Justify
your answer.
(4 marks)
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Continued
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(i)
Express 15 as a continued fraction, nd a convergent which diers from
15 by less than 10−5 and nd a solution of the Pell equation x2 − 15y 2 = 1 in positive
integers with x > 100.
(17 marks)
5
√
(ii) Dene the Fibonacci sequence (fn ) and prove by induction or otherwise
that, for all natural numbers n,
f1 − f2 + f3 − f4 + . . . − f2n + f2n+1 = 1 + f2n .
(8 marks)
End of Question Paper
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