Elements of Algebra

CHAP. IV. OF ALGEBRA. $11$ two or more numbers. It is also particularly worthy of observation, that if we write these numbers in succession as they follow each other, thus, $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, \&c. *$ we can trace no regular order; their increments being sometimes greater, sometimes less; and hitherto no one has been able to discover whether they follow any certain law or not. $41$. All composite numbers, which may be represented by factors, result from the prime numbers above mentioned; that is to say, all their factors are prime numbers. For, if we find a factor which is not a prime number, it may always be decomposed and represented by two or more prime numbers. When we have represented, for instance, the number * All the prime numbers from $1$ to $100000$ are to be found in the Tables of divisors, which I shall speak of in a succeeding note. But particular Tables of the prime numbers from $1$ to $101000$ have been published at Halle, by M. Kruger, in a German work entitled "Thoughts on Algebra;" M. Kruger had received them from a person called Peter Jaeger, who had calculated them. M. Lambert has continued these Tables as far as $102000$, and republished them in his supplements to the logarithmic and trigonometrical Tables, printed at Berlin in $1770$; a work which contains likewise several Tables that are of great use in the different branches of mathematics, and explanations which it would be too long to enumerate here. The Royal Parisian Academy of Sciences is in possession of Tables of prime numbers, presented to it by P. Mercastel de l'Oratoire, and by M. du Tour; but they have not been published. They are spoken of in Vol. V. of the Foreign Memoirs, with a reference to a memoir, contained in that volume, by M. Rallier des Ourmes, Honorary Counsellor of the Presidial Court at Rennes, in which the author explains an easy method of finding prime numbers. In the same volume we find another method by M. Rallier des Ourmes, which is entitled, "A new Method for Division, when the Dividend is a Multiple of the Divisor, and may therefore be divided without a remainder; and for the Extraction of Roots when the Power is perfect." This method, more curious, indeed, than useful, is almost totally different from the common one: it is very easy, and has this singularity, that, provided we know as many figures on the right of the dividend, or the power, as there are to be in the quotient, or the root, we may pass over the figures which precede them, and thus obtain the quotient. M. Rallier des Ourmes was led to this new method by reflecting on the numbers terminating the numerical expressions of products or powers, a species of numbers which I have remarked also, on other occasions, it would be useful to consider. F. T.