two expressions which are identical as to their form; and, therefore, every odd divisor of the formula ť2+3u' is also of the form yo + 3x2. Remark. With regard to the divisor 3, it is obvious that we must have y=0, and 2=1; or 3=0+3.1'; but for all other divisors this exception has not place. For example, 5 +3.6 = 133 =7.19; and 7=22+3, 12, also 19=42+3. 1', both of the same form. PROP. VIII. 109. Every odd divisor of the formula - 5u' is also of the same form yo — 522, For all its divisors are contained in the formula py2 + 2qyz — rz3, in which q= or < - pr− q2 = — a, or pr+q=5, and √ 5 ; and, consequently, q= 1 or 0; but the first case gives only even divisors, the same as in the foregoing proposition; and the latter caso. of q=0 reduces the above formula to 5y-2°, or y'-5%, which are identical forms; because and, consequently, every odd divisor of the formula t-5u is itself of the same form. As an example in this case, we may assume 9510-5. 1°, which is only divisible by 5 and 19. Now 5=5° - 5.2o, and 19=7-5.2°; and the same of all other numbers in the above form. Scholium. From the foregoing propositions it appears, that all numbers which are comprised in the following formulæ, viz. t tử, t+2, t° – 2ử, t +3u, and t° – 5u, t and u being prime to each other, can only have divisors that are of the same form. It is only necessary to except those divisors of the two latter forms, t + 3u3, and to — 5u3, that are double of an odd number; the reason for which exception is explained in arts. 108 and 109. It frequently happens, that a number falls under two or more of the above forms, in which case its divisors are also of the same double or treble forms. And in some cases we have numbers that belong to each of the forms above given. Thus 241 = 152 + 4* = 132 + 2.62 = 21a — 2,10° = 7° +3,82 = 312 - 5,122. 200 CHAP. IX. On the Quadratic Forms of Prime Numbers, with Rules for determining them in certain Cases. Lemma. 110. Since all square numbers are of one of the forms 4n, or 8n+1, we establish at once the three following theorems: 1. Every odd number represented by the formula y2+z2±4n+1. 2. Every odd number represented by the formula y2+2z8n+1, or 8n +3. 3. Every odd number represented by the formula y2 − 2x2±8n+1, or 8n+7. And from these three arise, by way of exclusion, three others; viz. 4. No number of the form 4n-1 can be represented by the formula y2+. 5. No number of the form 8n+5, or 8n+7, can be represented by the formula y2 + 2x2. 6. No number of the form 8n+3, or 8n+5, can be represented by the formula y2 - 22°. PROP. I. 111. Every prime number of the form 4n+ 1 is the sum of two squares, or is contained in the formula y2+22. For let m represent a prime number of this form, or m=4n+1; then (art. 87) (x-1)=м(m), or (x-1)=м(m). But x1 − 1 = (x2 + 1)(x2" — 1), and each of these factors has 2n values of a contained between the limits +1m and −1m, that render them divisible by m (cor. 1, art. 88), whence the factor +1 is divisible by m; but " + 1 is the sum of two squares, and therefore its divisor m is also the sum of two squares; because every divisor of the formula + u2 is itself of the same form (art. 105).-a. E. d. D. Cor. 1. As the form 4n+1 includes the two, 8n+1 and 8n+5; therefore, every prime number contained in these two latter forms is also the sum of two squares. Thus, 5, 13, 17, 29, 37, and 41, are prime numbers of the form 4n+1, and each of these is the sum of two squares; for 5=22 + 12, 13 = 3a + 2a, 17=42+1,29= 5° +2°, 37=6° + 12, and 41 = 52 + 42; and so on for all other prime numbers of this form. Cor. 2. We have seen (art. 91) that every number, which is produced from the multiplication of factors that are the sums of two squares, is itself of the same form, and may be resolved into two squares different ways, according to the number of its factors; and hence we may find a number, that is resolvible into two squares as many ways as we please, by multiplying together different prime numbers of the form 4n+1. PROP. II. 112. Every prime number 8n + 1 is of the three forms y + z2, y2 + 2x2, and y2 — 2x2. For let m be a prime number of the form 8n + 1, or m=8n+ 1; then, as this form is included in that of 4n + 1, we know, from the foregoing proposition, that my2+; and it therefore only remains to demonstrate the two latter cases. 2 Now since (1-1)=м(m), or (x-1)=м(m) (art. 87), we may put this under the form (x + 1)(x − 1); and each of these factors will have 4n values of x<m, that render them divisible by m (cor. 1, art. 88), so that there are so many different values of r, that render the binomial x + 1 divisible by m; but this may be put under the form (x2 - 1)2+ 2x2, and m being a divisor of this formula, it is itself of the same form y2 + 2x2 (art. 106). We may also put the same quantity "+1, under the form (x2 + 1)2 - 2x2; and m being also a divisor of this formula, is itself of the same form 32-222 (art. 107). Hence, every prime number of the form 8n+1 is of the three forins y2+x, y2+ 2x2, and y' — 2x2. a. E. D. Thus 41=5 +42=3°+2.4 = 73 - 2 . 2%, PROP. III. 113. Every prime number 82+3 is of the form y2+2x2. For let m be a prime number of this form, or m=8n+1; then we have (by art. 87) 2+2 (xTM - 1 — 1) = м(m), or (xn+2 − 1) = м(m); and there are 8n+2 values of a contained in the |