- #1
Math100
- 797
- 221
- Homework Statement
- In 1752, Goldbach submitted the following conjecture to Euler: Every odd integer can be written in the form ## p+2a^2 ##, where ## p ## is either a prime or ## 1 ## and ## a\geq 0 ##. Show that the integer ## 5777 ## refutes this conjecture.
- Relevant Equations
- None.
Proof:
Suppose ## 5777=p+2a^2 ##, where ## p ## is either a prime or ## 1 ## and ## a\geq 0 ##.
Now we consider two cases.
Case #1: Suppose ## p ## is a prime and ## a\geq 0 ##.
Let ## p=2 ##.
Then ## 5775=2a^2 ##.
Thus ## a=\pm \sqrt{2887.5} ##,
which contradicts the fact that ## a\geq 0 ##.
Case #2: Suppose ## p=1 ## and ## a\geq 0 ##.
Then ## 5776=2a^2 ##.
Thus ## a=\pm \sqrt{2888} ##,
which contradicts the fact that ## a\geq 0 ##.
Therefore, the integer ## 5777 ## refutes this conjecture.
Suppose ## 5777=p+2a^2 ##, where ## p ## is either a prime or ## 1 ## and ## a\geq 0 ##.
Now we consider two cases.
Case #1: Suppose ## p ## is a prime and ## a\geq 0 ##.
Let ## p=2 ##.
Then ## 5775=2a^2 ##.
Thus ## a=\pm \sqrt{2887.5} ##,
which contradicts the fact that ## a\geq 0 ##.
Case #2: Suppose ## p=1 ## and ## a\geq 0 ##.
Then ## 5776=2a^2 ##.
Thus ## a=\pm \sqrt{2888} ##,
which contradicts the fact that ## a\geq 0 ##.
Therefore, the integer ## 5777 ## refutes this conjecture.