Using the model to find the solution: It is a simplified representation of the actual situation It need not be complete or exact in all respects It concentrates on the most essential relationships and ignores the less essential ones. It is more easily understood than the empirical i. It can be used again and again for similar problems or can be modified.
Version for printing It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity.
It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked, see for example [ 17 ] where detailed justification for this idea is given.
Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties. Before we begin to look at the history of the study of perfect numbers, we define the concepts which are involved.
Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number. An aliquot part of a number is a proper quotient of the number. So for example the aliquot parts of 10 are 1, 2 and 5.
Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i. A perfect number is defined to be one which is equal to the sum of its aliquot parts. The four perfect numbers 6, 28, and seem to have been known from ancient times and there is no record of these discoveries.
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.
However, although numbers are represented by line segments and so have a geometrical appearance, there are significant number theory results in the Elements. The result which is if interest to us here is Proposition 36 of Book IX of the Elements which states [ 2 ]: Here 'double proportion' means that each number of the sequence is twice the preceding number.
Now Euclid gives a rigorous proof of the Proposition and we have the first significant result on perfect numbers. The Proposition now reads: The next significant study of perfect numbers was made by Nicomachus of Gerasa.
Around AD Nicomachus wrote his famous text Introductio Arithmetica which gives a classification of numbers based on the concept of perfect numbers. Nicomachus divides numbers into three classes: And those which are said to be opposite to each other, the superabundant and the deficient, are divided in their condition, which is inequality, into the too much and the too little.
However Nicomachus has more than number theory in mind for he goes on to show that he is thinking in moral terms in a way that might seem extraordinary to mathematicians today see [ 8 ], or [ 1 ] for a different translation: And in the case of those that are found between the too much and the too little, that is in equality, is produced virtue, just measure, propriety, beauty and things of that sort - of which the most exemplary form is that type of number which is called perfect.
Now satisfied with the moral considerations of numbers, Nicomachus goes on to provide biological analogies in which he describes superabundant numbers as being like an animal with see [ 8 ], or [ 1 ]: Deficient numbers are compared to animals with: Nicomachus goes on to describe certain results concerning perfect numbers.
All of these are given without any attempt at a proof. Let us state them in modern notation. We will see how these assertions have stood the test of time as we carry on with our discussions, but let us say at this point that assertions 1 and 3 are false while, as stated, 24 and 5 are still open questions.
However, since the time of Nicomachus we do know a lot more about his five assertions than the simplistic statement we have just made. Let us look in more detail at Nicomachus 's description of the algorithm to generate perfect numbers which is assertion 4 above see [ 8 ], or [ 1 ]: First set out in order the powers of two in a line, starting from unity, and proceeding as far as you wish: If, otherwise, it is composite and not prime, do not multiply it, but add on the next term, and again examine the result, and if it is composite leave it aside, without multiplying it, and add on the next term.
If, on the other hand, it is prime, and non-composite, you must multiply it by the last term taken for its composition, and the number that results will be perfect, and so on as far as infinity. As we have seen this algorithm is precisely that given by Euclid in the Elements.
However, it is probable that this methods of generating perfect numbers was part of the general mathematical tradition handed down from before Euclid 's time and continuing till Nicomachus wrote his treatise. Whether the five assertions of Nicomachus were based on any more than this algorithm and the fact the there were four perfect numbers known to him 6, 28, andit is impossible to say, but it does seem unlikely that anything more lies behind the unproved assertions.
Some of the assertions are made in this quote about perfect numbers which follows the description of the algorithm [ 1 ]: And it is their accompanying characteristic to end alternately in 6 or 8, and always to be even.
When these have been discovered, 6 among the units and 28 in the tens, you must do the same to fashion the next. Despite the fact that Nicomachus offered no justification of his assertions, they were taken as fact for many years.
Of course there was the religious significance that we have not mentioned yet, namely that 6 is the number of days taken by God to create the world, and it was believed that the number was chosen by him because it was perfect. Again God chose the next perfect number 28 for the number of days it takes the Moon to travel round the Earth.
Saint Augustine writes in his famous text The City of God: God created all things in six days because the number is perfect The Arab mathematicians were also fascinated by perfect numbers and Thabit ibn Qurra wrote the Treatise on amicable numbers in which he examined when numbers of the form 2np, where p is prime, can be perfect.Rawlings Velo Fastpitch Softball Bat oz FP8V11 30 inches, 19 oz.,new in wrapper, with receipt (emailed) The Velo FP Bat is for the female hitter looking for a ton of bat speed and incredible bat attheheels.com Rating: % positive.
Here's a chart with the primes and composites from attheheels.com is the most comprehensive prime numbers site with worksheets, review, quizzes, and other resources! 2, 3, 5, 7, 11, 13, 17, So a number is prime if it is a natural number-- and a natural number, once again, just as an example, these are like the numbers 1, 2, 3, so essentially the counting numbers starting at 1, or you could say the positive integers.
Whole Numbers: The numbers 1,2,3, are called natural numbers or counting numbers. Let us add one more number i.e.,zero(0), to the collection of natural numbers. Now the numbers are 0,1,2,. What is the ratio of number of primes and numbers of composites between 20 and 40?
The prime numbers between 20 and 40 are 23, 29, 31 and 37 that is 4 out of 20 or 1 to 5 Share to. Write down all the numbers that you have left. You should have 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, Check that these are the numbers that you have not crossed out.