A prime number is that which is measured by a unit alone. Perfect number, prime number, learning maths, mathematics. He began book vii of his elements by defining a number as a multitude composed of units. Euclid s elements is one of the most beautiful books in western thought. By contrast, euclid presented number theory without the flourishes. The only known perfect numbers are n nth perfect number number of digits. 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. Number theory euclids algorithm stanford university. The first recorded mathematical result concerning perfect numbers which is known occurs in euclid s elements written around 300bc. However, although numbers are represented by line segments and so have a geometrical appearance, there are. Book 10 proves the irrationality of the square roots of nonsquare integers e. As it appears in book ix, proposition 36 of his elements, euclid writes. It begins with the 22 definitions used throughout these books.
Then we will show that if nis an even perfect number then n 2 k1 2 1 where 2k 1 is a prime number. Book vii is the first of the three books on number theory. The proof of this theorem is broken up into two parts. So since 6 is a perfect multiple of 3, \\gcd6,3 3\, and we have found that \\gcd33,27 3\. Euclid was instrumental in the advances made in the study of perfect numbers. On the proof that all even perfect numbers are of euclids type. In it he claimed that euclids formula 2 k1 2 k 1 gives a perfect number for all odd integers k, see. Perfect numbers, mersenne primes, and the euclideuler theorem. Philo of alexandria in his firstcentury book on the creation mentions perfect numbers, claiming that the world was created.
Euclids elements, book vii department of mathematics. Aunitis that by virtue of which each of the things that exists is called one. First we will show that if n 2 k1 2 1 where 2k 1 is prime, then nis an even perfect number. 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. 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. 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. The important definitions are those for unit and number, part and multiple, even and odd, prime and relatively prime, proportion, and perfect number. Learning maths perfect number and prime number the statesman.
Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. In number theory, a perfect number is a positive integer that is equal to the sum of its positive. He later defined a prime as a number measured by a unit alone i. The greater number is amultipleof the lesser when it is measured by3 the lesser number. Each proposition falls out of the last in perfect logical progression. No particular kind of magnitude is specified in that book.1093 249 804 952 664 190 30 540 1485 277 102 722 755 402 524 479 286 1489 553 1298 496 626 593 196 747 583 1137 918 985 917 1352 1346 493 1223 1172 242 120 1047 594 147 1464 1177 1379 960