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Investigate prime numbers

1.Can you express every odd prime number as the difference of two squares?

4. Some square numbers can be formed by adding 2 prime numbers together.

Eg. 4=2+2 ; 9=7+2 : 16=11+5 and 16=13+3

Investigate whether it is possible to form every square number by adding two primes together.

5.What is Goldbach's Conjecture?

6. Is there a largest prime number? Try to find it or proof that it does not exist.

I don't understand most of the questions. All help will be appreciated. Ty

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1.Can you express every odd prime number as the difference of two squares?

This is an exercise in basic number theory.

Let p be an odd prime and $p = a^{2} - b^{2} = ( a + b ) ( a - b )$

Since p has no factors other than p and 1 then,

$p=a + b \;\ \text{and} \;\ a-b=1$

Therefore, $a=\frac{p+1}{2}$ and $b = \frac{p-1}{2}$

is the only way to express p as a difference of two squares.

$\left(\frac{p+1}{2}\right)^{2}-\left(\frac{p-1}{2}\right)^{2}=p$

This gives $p = a^{2} - b^{2}$ . QED

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5.What is Goldbach's Conjecture?

Every even number can be expressed as the sum of two primes.

Google this and you will find lots. It is still unproven.

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6. Is there a largest prime number? Try to find it or proof that it does not exist.

They are asking to prove the infinitude of the primes. The primes are infinite, therefore, there is no largest prime. Google it.

It goes back to Euclid. As a matter of fact, look up the GMPS (Great Mersenne Prime Search). The largest prime discovered to date, I believe, has over 10,000,000 digits.

For #4: think about it a little. If you find one that does not satisfy the condition, then it is disproved. How about 11^2=121? Can it be expressed as the sum of two primes?