You are Here: Home >< Maths

# soln of UKL society problems watch

1. 1) If x and y are prime numbers, solve the equation x^y - y^x = x.y^2 - 19.

The main tool is Fermat's little theorem: if p is prime then
a^p=a mod p. So (i)x divides y-19, and

(ii) y divides x+19, and in particular y<=x+19

Suppose y>19. By (i),x<=y-19 so by (ii) x=y-19. But
this is impossible (one of x,y would have to be even, hence x=2,y=21).

If y=19, then by (ii) x=19, which doesn't work.

So y<19 and by (i) x divides 19-y. This gives the
following possibilities:

y=2, x=17

y=3, x=2

y=5,x=2 or 7

y=7, x=2 or 3

y=11,x=2

y=13,x=2 or 3

y=17,x=2

Taking (ii) into account reduces the possibilities to three:

y=2,x=17

y=3,x=2

y=7,x=2
One checks by hand that the first doesn't work, but the other two do.
So there are exactly two solutions

2) find n so (1+4+9+16+....+n^2)/n is a square number

We want to make (n+1)(2n+1)/6 a square. First, this is not usually even an integer. As 2n+1 is odd, n+1 must be even, say n=2k+1. This gives (k+1)(4k+3)/3. So 3 must divide k or k+1.

Suppose k+1=3m. Then m(12m-1) is a square. As a prime p cannot divide both m and 12m-1, both m and 12m-1 must be squares, say m=r2, 12m-1=s2. Then 12r2=s2+1. Then s must be odd, say s=2p+1 and 6r2=2p2+2p+1. But the right-hand side is never divisible by 3 (consider mod 3). So this case is impossible.

Hence k=3m and (3m+1)(4m+1) must be a square. If a prime p divides 3m+1 and 4m+1, it divides m, so would not divide 3m+1. Hence 3m+1 and 4m+1 must both be squares, say 3m+1=p2, 4m+1=q2. Let q=p+r. Then m=2pr+r2 and p2-6pr-3r2-1=0. This gives p=
3r+√12r2+1 (the other solution is negative) and n=42r2+12r√(12r2+1)+1. As n increases with r, the smallest solution for n corresponds to the smallest value of r that makes this expression for n an integer. This value is r=2, giving n=337.
2. wow, Fermat and Galois all on one forum

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: June 29, 2004
Today on TSR

### Are you living on a tight budget at uni?

From budgets to cutbacks...

### University open days

1. University of Cambridge
Wed, 26 Sep '18
2. Norwich University of the Arts
Fri, 28 Sep '18
3. Edge Hill University
Faculty of Health and Social Care Undergraduate
Sat, 29 Sep '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams