The statement "Numbers that are 1 less than or 1 more than a multiple of 6 are…

Concept: Mathematical statements are classified by their logical status — by how they enter a mathematical system and what backs them. An axiom is accepted without proof as a starting assumption of the system. A theorem is a statement that has been established by a complete deductive proof from definitions, axioms and earlier results. A conjecture is a statement advanced as plausibly true on the strength of observed cases, before any proof settles it; a conjecture may later be proved, and it may equally be refuted by a single counterexample. Generalisation is the inductive step of extending a property observed in particular cases to a whole class — it describes how a claim is reached and, by itself, says nothing about whether that claim has been proved.

Application: The claim here is that every number one less than or one more than a multiple of 6 is prime. All that supports it is a handful of observed cases:

1. 6 − 1 = 5 and 6 + 1 = 7 — both prime.

2. 12 − 1 = 11 and 12 + 1 = 13 — both prime.

3. 18 − 1 = 17 and 18 + 1 = 19 — both prime.

No proof accompanies the claim, and it is not laid down as a starting assumption of the system. A claim offered on the strength of examples alone, ahead of any proof, is a conjecture.

Cross-check: Push the same pattern a little further and the claim fails:

• 24 + 1 = 25 = 5 × 5, which is composite.

• 36 − 1 = 35 = 5 × 7, which is composite.

So this conjecture is a false one — which is entirely possible: a conjecture is a proposal awaiting judgement, and a single counterexample is enough to refute it. What is true, and provable, is the converse: every prime greater than 3 has the form 6k − 1 or 6k + 1, because 6k, 6k + 2 and 6k + 4 are even while 6k + 3 is divisible by 3. The claim in the question reverses that theorem, and the reversal does not survive.

Contrast: Reading the four terms against this claim:

• Theorem — needs a completed deductive proof valid in every case; here a counterexample exists, so no such proof can exist.

• Axiom — needs the claim to be adopted without proof as a foundation on which the system is built; nothing of the kind is being assumed here.

• Generalisation — correctly describes how the claim was reached, by extending a pattern from a few examples; it is a description of the reasoning step and is silent about whether the resulting claim has been proved. Asked for the status of the statement itself, the term that names an unproved proposal awaiting proof or refutation is conjecture, and that is the answer recorded in the official key for this paper.

Hence the statement is a conjecture: a plausible-looking claim about primes advanced ahead of any proof — and, as the counterexamples show, one that turns out to be false.