He began book vii of his elements by defining a number as a multitude composed of units. However, although numbers are represented by line segments and so have a geometrical appearance, there are. No particular kind of magnitude is specified in that book. The greater number is amultipleof the lesser when it is measured by3 the lesser number. Perfect number, prime number, learning maths, mathematics. First we will show that if n 2 k1 2 1 where 2k 1 is prime, then nis an even perfect number. On the proof that all even perfect numbers are of euclids type. Book vii is the first of the three books on number theory.
Perfect numbers, mersenne primes, and the euclideuler theorem thomas browning may 2016 we say n is perfect when the sum of all of the factors of n equals 2n. It may come as a surprise to many people to learn that there are number theory results in euclids elements since it is thought of as a geometry book. Each proposition falls out of the last in perfect logical progression. It may come as a surprise to many people to learn that there are number theory results in euclid s elements since it is thought of as a geometry book. The first recorded mathematical result concerning perfect numbers which is known occurs in euclid s elements written around 300bc. The proof of this theorem is broken up into two parts. The only known perfect numbers are n nth perfect number number of digits. Euclids elements, book vii department of mathematics. Book 10 proves the irrationality of the square roots of nonsquare integers e.
In it he claimed that euclids formula 2 k1 2 k 1 gives a perfect number for all odd integers k, see. Aunitis that by virtue of which each of the things that exists is called one. In number theory, a perfect number is a positive integer that is equal to the sum of its positive. Euclid was instrumental in the advances made in the study of perfect numbers. As it appears in book ix, proposition 36 of his elements, euclid writes. The important definitions are those for unit and number, part and multiple, even and odd, prime and relatively prime, proportion, and perfect number. It may come as a surprise that ratios do not themselves form a kind of magnitude since they can be compared, but they cannot be added. He later defined a prime as a number measured by a unit alone i. A prime number is that which is measured by a unit alone.
Learning maths perfect number and prime number the statesman. Philo of alexandria in his firstcentury book on the creation mentions perfect numbers, claiming that the world was created. Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. So since 6 is a perfect multiple of 3, \\gcd6,3 3\, and we have found that \\gcd33,27 3\. Yet, rather remarkably, although unknown until comparatively recently, progress had been made.
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