Math1061/7861 Integers Worksheet With Answers Page 2
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1
23. (4 marks)
Prove that there exists a
prime number of the form n +2n 3, where n is a positive
integer.
Hint: (1) Find some value of n for which the expression is prime. Then you must show uniqueness,
so suppose there are two different values of n yielding primes p and q; then show that in fact we must
have p = q.
(2) Think about primes and factors; what can you do with a quadratic expression like n + 2n
3 ?
:
2
First note that n
+ 2n
3 = (n + 3)(n
1).
When n = 2, this equals 5, which is a prime.
2
So there exists at least one value of n for which n
+ 2n
3 is prime.
2
To show uniqueness, suppose there are two values of n which make n
+ 2n
3 prime. So
suppose we have
2
2
p = n
+ 2n
3 = (n
+ 3)(n
1)
and
q = n
+ 2n
3 = (n
+ 3)(n
1).
1
1
1
2
2
2
1
2
Now the only factors a prime number can have are 1 and the prime itself. And of course n + 3
2
and n
1 are both factors of n
+ 2n
3. Also n is a positive integer; so n + 3 must be 4 or
more. Therefore we must have (n
1) = 1 and (n
1) = 1, so that n
= n
= 2, so p = q
1
2
1
2
after all, and they both equal 5.
2
+
Hence there is a unique prime number of the form n
+ 2n
3 where n
.
Z
4. (6 marks)
Determine whether or not there exist solutions to the following linear Diophantine equa-
tions. If a solution exists, give one, that is, give integer values for x and y which satisfy the given
equation.
(a) 14x + 3003y = 7.
: Since 3003 = 7
429, and 14 = 7
2, a solution does exist.
For simplicity, cancel 7 throughout, and work with 2x + 429y = 1.
429 = 2
214 + 1.
So 1 = 429
1 + 2.( 214) = 2x + 429y.
Thus x =
214 and y = 1 are a solution to the equation.
(b) 18x + 1028y = 3.
: Here although 3 18, we have 3 1028, so there is no solution.
(c) 221x
255y = 17.
: Now 221 = 17
13 and 255 = 17
15, so a solution exists.
We cancel 17 throughout, and work with the equation 13x
15y = 1.
15 = 13
1 + 2;
13 = 2
6 + 1.
Thus 1 = 13
2
6 = 13
1
(15
13).6, or
1 = 13
7
15
6.
So x = 7 and y = 6 is a solution to 13x
15y = 1, and hence also to 221x
255y = 17.
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