Name: Tarrasita
Who is asking: Student
Level: Secondary

Hi.....

I was given some homework to investigate the following conjectures on prime numbers.I'm supposed to find patterns and formulae in them.And I have worked out as below:

1) There is at least one prime between consecutive square numbers.

 
Square number	Number of Primes

    1
       -------->	2
    4
       -------->	2
    9
       -------->	2
   16
       -------->	3
   25
   

2) Every prime except 2 and 3 is of the form 6n+1 where n is a natural number.

- 6(1)-1 = 5
6(1)+1 = 7

6(2)-1 = 11
6(2)+1 = 13

6(3)-1 = 17
6(3)+1 = 19

6(4)-1 = 23

6(5)-1 = 29
6(5)+1 = 31

3) Any odd prime which is of the form 4n+1 is equal to the sum of two perfect squares.

4(1)+1 = 5 = 1² + 2²
4(3)+1 = 13 = 2² + 3²
4(4)+1 = 17 = 1² + 4²
4(7)+1 = 29 = 2² + 5²
4(9)+1 = 37 = 1² + 6²
4(10)+1 = 41 = 4² + 5²

I've done the working, but couldn't find any pattern or formula. I know I've put too many questions in one letter but it's pretty urgent. I'd really appreciate if you could help me. I don't want to know the whole answer, just the hints to help me see. Thanx for your time.

I don't think that you are expected to find any pattern or formula. These are conjectures, that is statements that someone thought was true. I think that you are expected to show that the statement is true, show that the statement is false or give your opinion, after some experimentation, as to whether the statement is true or false.

I think you have already settled conjecture 2, that "Every prime except 2 and 3 is of the form 6n+1 where n is a natural number". In your list you didn't mention 6(4)+1 since 25 is not a prime. Thus the statement "Every prime except 2 and 3 is of the form 6n+1 where n is a natural number" is false when n is 4. Thus the conjecture is false.

For conjecture 3 that "Any odd prime which is of the form 4n+1 is equal to the sum of two perfect squares" I suggest that you keep going. 4(11)+1 and 4(12)+1 are not primes but 4(13)+1 and 4(14)+1 are primes. Is each equal to the sum of two perfect squares?

What about conjecture 1. Do you think it is true or false?