- #1

Math100

- 676

- 180

- Homework Statement:
- Give an example to show that the following conjecture is not true: Every positive integer can be written in the form p+a^2, where p is either a prime or 1, and a##\geq##0.

- Relevant Equations:
- None.

Disproof: Here is a counterexample:

Suppose p+a^2=25, where 25 is a positive integer.

Then we have 25=0+25

=9+16

=16+9

=21+4.

Note that none of 0, 4, 9, and 16 is prime or 1.

Therefore, not every positive integer can be written in the form p+a^2,

where p is either a prime or 1, and a##\geq##0.

Above is my proof/answer for this problem. Can anyone please review/verify if it's correct?

Suppose p+a^2=25, where 25 is a positive integer.

Then we have 25=0+25

=9+16

=16+9

=21+4.

Note that none of 0, 4, 9, and 16 is prime or 1.

Therefore, not every positive integer can be written in the form p+a^2,

where p is either a prime or 1, and a##\geq##0.

Above is my proof/answer for this problem. Can anyone please review/verify if it's correct?