- #1

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- Homework Statement
- If ## 1 ## is added to a product of twin primes, prove that a perfect square is always obtained.

- Relevant Equations
- None.

Proof:

Suppose ## p ## and ## p+2 ## are twin primes.

Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##.

Thus, ## (p+1)^2 ## is a perfect square.

Therefore, if ## 1 ## is added to a product of twin primes,

then a perfect square is always obtained.

Suppose ## p ## and ## p+2 ## are twin primes.

Then we have ## p(p+2)+1=p^2+2p+1=(p+1)^2 ##.

Thus, ## (p+1)^2 ## is a perfect square.

Therefore, if ## 1 ## is added to a product of twin primes,

then a perfect square is always obtained.