This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1844 Excerpt: ...if we take x = 1, 2, 4, 6, 8, 16, &c.; but Euler shows that if x = 32, then 2" + 1 = 641 x 6700417. It is manifest that the number of primes is infinite, since the series l + 4 + J + +-t--fa, &c. is infinite. Though we cannot find an algebraic formula which contains only primes, nor one which contains all such numbers, ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1844 Excerpt: ...if we take x = 1, 2, 4, 6, 8, 16, &c.; but Euler shows that if x = 32, then 2" + 1 = 641 x 6700417. It is manifest that the number of primes is infinite, since the series l + 4 + J + +-t--fa, &c. is infinite. Though we cannot find an algebraic formula which contains only primes, nor one which contains all such numbers, or is their general law, yet we have some formulae which are remarkable for the multitude of primes which they contain. Thus, if in the formula x2--x + 41, we take x = 0, 1, 2, 3, 4, &c. we shall have 41, 43, 47, 53, &c. of which the first 41 terms are primes; in the formula x2 + x+ 17 the first 17 terms are primes; and in the formula 2z2 + 29, the first 29 terms are primes. The expression 2a;+ 1, according as x is even or odd, contains the two forms 4 + 1 and 4a;--1, two principal divisions of prime numbers; the form 4a;+1 may be subdivided into 8a;-f-1 and 8a;--3, and the form 4a;--1, into 8a;--1 and 8a; + 3, so that all the primes may be reduced to these four forms. The forms 6z + 1 according as x is even or odd, give, in reference to 12, the four forms 12a;+ 1, 12a;-f-5, 12a;--5, and 12a;--1, each of which contains an infinite number of primes. In general, if a is any number, every odd number may be represented by 4aa; ziz b in which b is odd, and less than 2a, and if from among all the possible values of b we expunge those which have a common measure with a, the remaining 4aa;; b will contain all the prime numbers, divided in respect of the multiplier of 4a, into as many forms as b shall have different values, and these forms altogether contain the whole of the prime numbers. Suppose--2a + b, --a + b, b, a + b, ta + b, &c. to be an arithmetical progression, of which the general term is ax' + 6, ...
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