Question

Suppose an integer x a natural number n and a prime number p satisfy the equation 7x² – 44x + 12 = pⁿ. Find the largest value of p.

Answer 1

7x² – 44x + 12 = pⁿ

⇒ 7x² – 42x – 2x + 12 = pⁿ

⇒ (7x – 2)(x – 6) = pⁿ

Let 7x – 2 be the prime number

⇒ x – 6 must be the same prime number or 1

Now, x – 6 = 1

⇒ x = 7

Then, 7x – 2 = 7 × 7 – 2

= 47, which is a prime number

Further, 7x – 2 and x – 6 must be the same prime numbers

∴ 7x – 2 = x – 6

⇒ 6x = – 4

⇒ x = –2/3,

Hence, x ∉ Z

∴ pⁿ = 47

p = 47 and n = 1