" PROP. XIV. 157. If a be a prime number of the form 8n - 1, the equation x2 - ay2 = 2 is always resolvible in integers, For let p and q be the least numbers that establish the equation p2 - aq2 = 1, then we may have either q=mn, or q=2mn, according as we suppose q to be odd or even, which give the four following resolutions of the equation 1. p2 - 1= aq2; viz, Sp+1= am2, 3. { p + 1 = 2am2, The first of which forms gives am2 - n2 = 2, an equation that cannot obtain, because, m and n being both odd, the first side is of the form (8n-1)(8n'+1) - (8n" +1)=8n"" — 2, which can never be equal to 2, The third form gives am-n2=1, which is also impossible; for if m and n were both odd, then am2 - n' would be even, and, therefore, not equal to 1. If m was even and n odd, then am2 - n2±4n' + 3, which cannot be equal to 1; and we have the same result by taking n even and m odd: therefore, this equation is impossible. The fourth form'gives m2- an2 = 1, which cannot have place; because p and q are the least numbers that satisfy the equation p2 - aq=1. Therefore, the second is the only possible form, and this gives m2 — an2 = 2; and, consequently, the proposed equation x2 - ay2 = 2 is always possible, when a is a prime of the form 8n-1. 158. Cor. If, in the equation p2- aq=1, we resolve a into any two factors prime to each other, as mn, we have, by transposition," p2 - 1 = mnq, which equation may be decomposed into factors four different ways; viz. where f must be either 1 or 2, which numbers, being successively substituted for f, give the following eight combinations; viz. but of these, the seventh, viz. gmnh=1, cannot have place; because we suppose here, as in the foregoing proposition, that p and q are the least values that satisfy the equation p-aq=1, or p2 — mnq2 = 1. Now by means of these decompositions we readily draw the following conclusions: 1. If the numbers m and n are both of the form 4n+3, no one of the bottom equations can obtain; for, in this case, whatever forms we give to the two squares g and h, the equations will be of one of the forins 4n, 4n + 1, 4n + 3, no one of which can be equal to 2. The fifth equation is also impossible on the same supposition; because this, by transpoh2 + 1 sition, gives =g, an integer, whereas we have shown, that no number that is the sum of two squares prime to each other, can be divided by numbers of the form 4n+ 3 (art. 105, and lemma 4, page 200). There remains, then, only the two equations 1 and 3, one of which must, therefore, necessarily obtain; and hence we draw the following remarkable theorems. 1. If m and n be both of the form 41-3, the equation mx2 — ny2 = ± 1 will be always possible in integer numbers; that is, under one or other of the signs + or -1. If we suppose m and n to be both of the form 4n+1, then the same reasoning will apply, except to the fifth equation, and, therefore, in this case, our theorem must be expressed thus: 2. If m and n be both of the form 4n+1, then one of the equations x2 — mny2 = − 1, or mx2 - ny2 = ± 1, will always be resolvible in integers. And in a similar manner we may deduce the following theorem, which is still more general. 3. If m and m' be two prime numbers of the form 4n+3, and ʼn a prime number of the form 4n+1, it will be always possible to satisfy one of the three following equations: 317 CHAP. II. On the Solution of Indeterminate Equations of the First Degree. PROP. I. 159. To find the values of a and y in the equa tion ar-by= ± 1. We have already considered this equation (art. 141), and it is only repeated here to preserve uniformity, and to offer a few remarks that could not be properly introduced in that article. First, it may be observed, that a and b must be prime to each other, for otherwise the equation will be impossible; because the first side of the equation would be divisible by the common divisor of a and b, but the other side + 1 would not. But, if a and b have these conditions, then the equation is always possible in integer numbers. secutive terms of a series of converging fractions, then p°q-q°p=±1; and, therefore, to find the values of x and y, in the above equation, we have a only to convert into a series of converging frac tions, and to assume, for these quantities, the terms |