TMUA Official 2022 Paper 2 · Question 7

Consider the following claim:

The difference between two consecutive positive cube numbers is always prime.

Here is an attempted proof of this claim:

I. $(x+1)^3=x^3+3x^2+3x+1$
II. Taking $x$ to be a positive integer, the difference between two consecutive cube numbers can be expressed as $(x+1)^3-x^3=3x^2+3x+1$ .
III. It is impossible to factorise $3x^2+3x+1$ into two linear factors with integer coefficients because its discriminant is negative.
IV. Therefore for every positive integer value of $x$ the integer $3x^2+3x+1$ cannot be factorised.
V. Hence, the difference between two consecutive cube numbers will always be prime.

Which of the following best describes this proof?

AThe proof is completely correct, and the claim is true.

BThe proof is completely correct, but there are counterexamples to the claim.

CThe proof is wrong, and the first error occurs on line I.

DThe proof is wrong, and the first error occurs on line II.

EThe proof is wrong, and the first error occurs on line III.

FThe proof is wrong, and the first error occurs on line IV.

GThe proof is wrong, and the first error occurs on line V.

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