Which of the following is(are) true if p and q are primes and p>q (I)p^2-q^2 could be a prime (II) p^3-q^3 could be a prime (III) p^4-q^4 could be a primeĀ 



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Posted on (Answer #1)

It is given that p and q are prime numbers and p > q. A prime number X has only two positive divisors, 1 and X factors 1 and the number itself.

p^2 - q^2 = (p - q)(p + q): The number p^2 - q^2 could be a prime number if p - q = 1

p^3 - q^3 = (p - q)(p^2 + pq + q^2): This could also be a prime number if p - q = 1

p^4 - q^4 = (p - q)(p + q)(p^2 + q^2): This number has 4 positive divisors, 1,(p - q),(p + q) and (p^2 + q^2). Even if p - q = 1, there are two other divisors of the number.

Statements (I) and (II) are true, (III) is false.

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