Elements of Algebra

CHAP. VII. OF ALGEBRA. 351 CHAP. VII. *Of a particular Method, by which the Formula $an^2 + 1$ becomes a Square in Integers.* 96. That which has been taught in the last chapter, can-not be completely performed, unless we are able to assign for any number $a$, a number $n$, such, that $an^2 + 1$ may become a square; or that we may have $m^2 = an^2 + 1$. This equation would be easy to resolve, if we were satisfied with fractional numbers, since we should have only to make $m = 1 + \frac{np}{q}$; for, by this supposition, we have $m^2 = 1 + \frac{2np}{q} + \frac{n^2p^2}{q^2} = an^2 + 1$; in which equation, we may expunge $1$ from both sides, and divide the other terms by $n$: then multiplying by $q^2$, we obtain $2pq + np^2 = anq^2$; and this equation, giving $n = \frac{2pq}{aq^2 - p^2}$, would furnish an infinite number of values for $n$: but as $n$ must be an integer number, this method will be of no use; and therefore very different means must be employed in order to accomplish our object. 97. We must begin with observing, that if we wished to have $an^2 + 1$ a square, in integer numbers, (whatever be the value of $a$), the thing required would not be possible. For, in the first place, it is necessary to exclude all the cases, in which $a$ would be negative; next, we must exclude those also, in which $a$ would be itself a square; because then $an^2$ would be a square, and no square can become a square, in integer numbers, by being increased by unity. We are obliged, therefore, to restrict our formula to the condition, that $a$ be neither negative, nor a square; but whenever $a$ is a positive number, without being a square, it is possible to assign such an integer value of $n$, that $an^2 + 1$ may become a square: and when one such value has been found, it will be easy to deduce from it an infinite number of others, as was taught in the last chapter: but, for our purpose, it is sufficient to know a single one, even the least;