Elements of Algebra

CHAP. XIII. OF ALGEBRA. 405 such as $x^2 + y^2$, may be also a fifth power, we shall have $a = 1$, and $c = 1$; therefore, $x = p^5 - 10p^3q^2 + 5pq^4$; and $y = 5p^4q - 10p^2q^3 + q^5$; and, farther, making $p = 2$, and $q = 1$, we shall find $x = 38$, and $y = 41$; consequently, $x^2 + y^2 = 3125 = 5^5$. CHAP. XIII. *Of some Expressions of the Form $ax^4 + by^4$, which are not reducible to Squares.* 202. Much labour has been formerly employed by some mathematicians to find two biquadrates, whose sum or difference might be a square, but in vain; and at length it has been demonstrated, that neither the formula $x^4 + y^4$, nor the formula $x^4 - y^4$, can become a square, except in these evident cases; first, when $x = 0$, or $y = 0$, and, secondly, when $y = x$. This circumstance is the more remarkable, because it has been seen, that we can find an infinite number of answers, when the question involves only simple squares. 203. We shall give the demonstration to which we have just alluded; and, in order to proceed regularly, we shall previously observe, that the two numbers $x$ and $y$ may be considered as prime to each other: for, if these numbers had a common divisor, so that we could make $x = dp$, and $y = dq$, our formulae would become $d^4p^4 + d^4q^4$, and $d^4p^4 - d^4q^4$: which formulae, if they were squares, would remain squares after being divided by $d^4$; therefore, the formulae $p^4 + q^4$, and $p^4 - q^4$, also, in which $p$ and $q$ have no longer any common divisor, would be squares; consequently, it will be sufficient to prove, that our formulae cannot become squares in the case of $x$ and $y$ being prime to each other, and our demonstration will, consequently, extend to all the cases, in which $x$ and $y$ have common divisors. 204. We shall begin, therefore, with the sum of two biquadrates; that is, with the formula $x^4 + y^4$, considering $x$ and $y$ as numbers that are prime to each other: and we have to prove, that this formula becomes a square only in the cases above-mentioned; in order to which, we shall enter