Elements of Algebra

392 ELEMENTS PART II. together an even and an odd square, the sum will always have the form of $4n+1$, and never of $4n-1$. Farther, every prime number, which belongs to the formula $4n+1$, is the sum of two squares; this is undoubtedly true, but it is not easy to demonstrate it*. 173. Let us proceed farther, and consider the formula $x^2+2y^2$, that we may see what values we must give to $x$ and $y$, in order that it may have factors. As this formula is expressed by the imaginary factors $(x+y\sqrt{-2}) \times (x-y\sqrt{-2})$, it is evident, as before, that, if it have divisors, these imaginary factors must likewise have divisors. Suppose, therefore, $x+y\sqrt{-2} = (p+q\sqrt{-2}) \times (r+s\sqrt{-2})$, whence it immediately follows, that $x-y\sqrt{-2} = (p-q\sqrt{-2}) \times (r-s\sqrt{-2})$, and we shall have $x^2+2y^2 = (p^2+2q^2) \times (r^2+2s^2)$; so that this formula has two factors, both of which have the same form. But it remains to determine the values of $x$ and $y$, which produce this transformation. For this purpose, we shall consider that, since $x+y\sqrt{-2} = pr - 2qs + qr\sqrt{-2} + ps\sqrt{-2}$, and $x-y\sqrt{-2} = pr - 2qs - qr\sqrt{-2} - ps\sqrt{-2}$, we have the sum $2x = 2pr - 4qs$; and, consequently, $x = pr - 2qs$: also the difference $2y\sqrt{-2} = 2qr\sqrt{-2} + 2ps\sqrt{-2}$; so that $y = qr+ps$. When, therefore, our formula $x^2+2y^2$ has factors, they will always be numbers of the same kind as the formula; that is to say, one will have the form $p^2+2q^2$, and the other the form $r^2+2s^2$; and, in order that this may be the case, $x$ and $y$ may also be determined in two different ways, because $q$ may be either positive or negative; for, we shall first have $x = pr - 2qs$, and $y = ps + qr$; and, in the second place, $x = pr + 2qs$, and $y = ps - qr$. 174. This formula $x^2+2y^2$ comprehends therefore all the numbers which result from adding together a square and twice another square. The following is an enumeration of these numbers as far as 50; 1, 2, 3, 4, 6, 8, 9, 11, 12, 16, 17, 18, 19, 22, 24, 25, 27, 32, 33, 34, 36, 38, 41, 43, 44, 49, 50. * The curious reader may see it demonstrated by Gauss, in his "Disquisitiones Arithmeticæ;" and by De la Grange, in the Memoirs of Berlin, 1768.