Perfect Numbers - KS3

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A number is perfect if it is equal to (or less or more) than the sum of its aliquot parts (its factors). For example,

6 is perfect as 6=1+2+3;

8 is deficient as 8>1+2+4 or

12 is abundant as 12=1+2+3+4+6.

Some of the discovered perfect numbers are the following:

• Nicomachus 6, 28, 496, 8128 (they all ended in 6 or 8).
• Theon of Smyrna (c. 125 AD) 6 and 28 only.
• Iamblichus (c. 325 AD) asserted that there is only one perfect number in each of the intervals 1, 10, 100, 1000, 10 000 etc., and the perfect numbers end alternately in 6 and 8. We now know that both of these statements are UNTRUE but you can find them repeated in the arithmetic of Boethius c. 510. It is believed that the authors from the Middle Ages and the Renaissance followed these two errors.

The fifth perfect number is given as 33 550 336 in an anonymous manuscript in 1456-1461.

The Pythagoreans used this term in another sense, because apparently 10 was considered by them to be a perfect number.

What people thought of primes through the history

Prime number (as the one defined by Aristotle, Euclid and Theon of Smyrna) is a number "measured by no number but by an unit alone" Iambilicus said that a prime number is also called "odd times odd".

Prime number was apparently first described by Pythagoras.

Iamblichus writes that Thymaridas called a prime number rectilinear since it can only be represented one-dimensionally.

In English prime number is found in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

Some older textbooks include 1 as a prime number.

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