Final answer:
The statement that every odd integer is the difference of two squares is a mathematical claim that can be directly proved using basic number theory principles, without the need for large theorems like Fermat's Last Theorem or Goldbach's Conjecture. One can construct such a proof by representing an odd integer as '2m + 1' and expressing it as the difference between the squares '(m+1)^2' and 'm^2'. Hence, none of the options listed completely fits as an answer, but Proof by contradiction (C) is the closest match in context, although the actual proof method is direct construction.
Step-by-step explanation:
The question deals with the assertion that every odd integer is the difference of two squares. Obviously, this is a claim in the realm of number theory, a branch of pure mathematics. To address this, we do not need large theorems such as Fermat's Last Theorem or the complicated Goldbach's Conjecture. We can provide a simple proof by construction for odd integers.
Let's say 'n' is an odd integer. Since 'n' is odd, it can be expressed as 'n = 2m + 1' for some integer 'm'. We can then choose two squares, '(m+1)^2' and 'm^2'. Subtracting these, we get '(m+1)^2 - m^2 = (m^2 + 2m + 1) - m^2 = 2m + 1 = n'. Hence, 'n' is expressed as the difference of two squares.
The correct option to choose from the given choices would be C) Proof by contradiction. However, the actual answer is not strictly a proof by contradiction but a direct proof by construction. The point of the question seems to be recognizing the basic principles of number theory in action, rather than applying any of the specific theorems or conjectures listed in the choices.