How to Write Proofs

1. Prove that

2. Given a set of two million points on the plane. Prove that there exists a
straight line that separates this set into two subsets of exactly one million
points each.

3. (Fermat's Little Theorem) If a natural number a is not divisible by a prime
number p, that is (mod p).

4. There exist arbitrarily large gaps between consecutive prime numbers.

5. (Chinese Remainder Theorem)
Consider the system of k linear congruences. (mod m_i)
(i = 1, 2, 3, ..., k), where (m_i,m_k) = 1 for all i ≠ j. Let m = m₁m₂m₃ ...m_k.
This system has a unique solution x ₀ (mod m).

6. Prove there are in nitely many prime numbers.

7. A Fermat prime is a prime of the form . How many Fermat primes
are there?

8. Find a formula P (n) so that it produces distinct primes for each natural
number n.

9. (Goldbach Conjecture) Is every even number greater than two a sum of
two primes ?

10. Twin primes are primes that differ by two. Are there infinitely many twin
primes?

11. A regular p-gon, where p is prime, is constructible if and only if p is a
Fermat prime.

12. Prove that is not a rational number .

"PROOFS" Problem Set #1

1. There are infinitely many primes of the form 6n + 5.

2. If (a,m) = 1, then (mod m) has a unique solution.

3. If is prime then n is prime.

4. Given any three odd positive integers, it is possible to find a fourth odd
positive integer such that the sum of the squares of all four numbers is a
perfect square.

5. The sequence 2ⁿ-3, n = 2, 3, 4, ... contains infinitely many terms divisible
by 5 and infinitely many terms divisible by 13, but no terms divisible by
(5)(13).

6. Is it possible for a sum of reciprocals of distinct primes to add up to an
integer?

7. Is it possible for a sum of reciprocals of the first n, n > 1, natural numbers
to add up to an integer?

8. Factor a ⁴ + 4b⁴.

9. For natural numbers n, when is n⁴ + 4ⁿ prime?

10. Is 4⁵⁴⁵ + 545⁴ prime?

11. Prove or disprove that there are two rational numbers such that
p + q = pq = 1

12. Prove or disprove that there are three rational numbers such that p+q+r =
pqr = 1.

13. Prove or disprove that there are four rational numbers such that p + q +
r + s = pqrs = 1.

14. If the sum of two positive integers is 2310, then their product is not divisible by 2310.

15. Let a, b, c and p be real numbers, with a, b, c not all equal, such that
Determine all possible values of p and prove
that abc + p = 0.

16. If all of the positive integers are separated into two subsets with no common
elements, then one of these two subsets must contain a three term
arithmetic progression .

17. Find all non negative integers x, y satisfying

18. Find all prime numbers p such that is also a prime.

19. Prove or disprove that there is a positive integer n such that is a rational number.

20. Find the smallest integer which can be written as the sum of nine, the
sum of ten and the sum of eleven consecutive positive integers.

21. 4ⁿ + 2 is divisible by 6 for every positive integer n.

22. If a, b, c are positive numbers such that abc = 1, then

23. If in a set of 21 numbers, the sum of any 10 is less than the sum of the
other 11, then all of the numbers are positive.

24. The closed curve ABCDEFGHA consists of 8 straight line segments
AB,BC,CD,DE,EF, FG,GH and HA. The 8 points A,B,C,D,E, F,G
and H lie at the vertices of a cube. The curve does not intersect itself.
Then at least one of the segments of the curve coincides with an edge of
the cube.

25. If quadrilateral ABCD is inscribed in a circle with center O and ,
then the area of AOCB = the area of AOCD.

26. Triangle ABC is equilateral and P is a point in the interior of the triangle
such that the perpendicular distances to the sides, PQ = 6, PR = 8, and
PS = 10. Find the area of the triangle.