powers from 2 to 12, the 7th powers only excepted, which cannot be introduced into these forms, because neither 7+1, nor 2.7 +1, is a prime number. 6 Scholium. By means of the foregoing table of formulæ, we may frequently satisfy ourselves of the possibility or impossibility of equations of the form a.x" ± by" = dz". And also whether any given number a is a complete power or not, without the trouble of extracting its root: it is to be observed, however, that a given number may be of a possible form, though it be not a complete power; but if it be of an impossible form, then we are certain, without any farther trouble, that it is not a complete power. PROP. VIII. ss. If m be a prime number, and P be made to represent any polynomial of the nth degree, as then, I say, there cannot be more than n values of x, between the limits +m, and m, that render this polynomial divisible by m. For let k be the first value of x, that renders P divisible by m, so that Am = 7-1 -3 k" + ak" ' + bk2¬2 + ck2¬3 + then, by subtraction, we have P— Am = (x” — k") + a{x" ~ ' — k" − 1) + b (x” ~ 2 — k" −2) + &c. But the latter side of this equation, being divided by x-k (art. 80), we shall have for a quotient a polynomial of the degree n-1; which, being represented by p', gives p — sm = (x — k)p′, or p= (x − k)p' + am. Let now k' be a second value of x, that renders P divisible by m, then it follows, that (x−k)P' + am is also divisible by m; and, consequently, (x-k)p divisible by m, but the factor x-k, which now becomes (k' — k), cannot be divisible by m, because both and k are less than 4m; therefore, P cannot be divisible a second time by m, unless p' be divisible by m. The polynomial P is, therefore, only once more divisible by m than the polynomial p'; and, in the it may be shown, that P', of the degreen-1, is only once more divisible by m, than same manner, p", of the n-2 degree, &c.; and hence it follows, that, r being a polynomial of the n degree, there can be only n different values of æ, comprised between the limits +m and m, that render it divisible by m. -- Q. E. D. Cor. We have seen (cor. 1, art. 87), that if m be a prime number, the formula a-1-1 has m-1 values of x, between the limits +m and -4m, that render it divisible by m. Now, this being put under the formn (+1)*(-1), it follows, that ર the limits +m and m, that render them diFor neither of them can have more visible by m. than m 1 2 such values, by the foregoing propo sition; and, since their product has m-1, it is obvious, that they have each the same number of values of x between the above limits, and that this m-1 number is 2 176 CHAP. VII. On the Products and Transformations of certain Algebraical Formulæ. PROP. I. 89. The product of the sum and difference of any two quantities is equal to the difference of their squares. For, (x + y)(x − y) = x2 — y3. PROP. II. Q. E. D. 90. The product of a sum of two squares, by double a square, is also the sum of two squares; or (x2 + y2) × 2z2#x22 + y22. For, (x2 + y2) × 22' = (x + y)2. z2 + (x − y)2. ≈2, which is evidently "+y". Cor. Hence if a number be the sum of two squares, its double is also the sum of two squares. Also if a number N be the sum of two squares, 2′′N is so likewise. Thus, for example, 5=2+1; 5x2=10=3°+1; 10 × 220=4° +2°; 20 x 240=6° + 2o &c. PROP. III. 91. The product arising from the sum of two squares by the sum of two squares, is also the sum of two squares; or For, (x2+y°)(x22+y^2)=x"2+y′′2. (x2 + y') (x^ + y^) = { (xx' + yy')' + (xy' — x'y)', or (xx' — yy')2 + (xy' + x'y)3, as will be evident from the development of these expressions; and, consequently, (x2+y°)(x22 +y'2)±x”2+y′′. Q. E. D. Cor. Hence the product may be divided into two squares two different ways. And if this product be again inultiplied by another, that is the sum of two squares, the resulting product may be divided into two squares four different ways; and, generally, if a number N be the product of n factors, each of which is the sum of two squares, then will N be the sum of two squares, and may be resolved into two squares 2" different ways. Then the product 6582 + 1, or 7* + 4°. Again, The product { 17=42+12 = 1105322+9° 33° +4° 31°+12° =242 +23°. And this resolution of the given product into square parts, is readily effected by the foregoing theorem; for ƒ (82 + 1 ) (42 + 1o) = (4 . 8 + 1) + (8.1-4.1)'= (4.8-1)2+(8.1 +4. 1), and = ƒ (7° +4°)(4° + 1) = (4 . 7 + 1 . 4)2 + (4.4 − 7.1)2 = (4.7 - 1.4)2 + (7 . 1 + 4 . 4)2. { And in the same manner may any other product, arising from factors of this form, be resolved into its square parts: |