Perfect numbers
Mathematicians have been fascinated for millenniums by the properties and patterns of numbers. They have noticed that some numbers are equal to the sum of all of their factors (not including the number itself). Such numbers are called perfect numbers. A perfect number is a whole number, an integer greater than zero and is the sum of its proper positive devisors, that is, the sum of the positive divisors excluding the number itself. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors, including itself. For example; 6, the first perfect number, its factors are 1, 2, 3 and 6. So, 1 + 2 + 3 = 6 or alternatively (1+2+3+6) divided by 2, which = 6. The next perfect number is 28, and its factors are 1, 2, 4, 7, 14, and 28. So, if we use the method which excludes the perfect number 28, we express the equation as, 1+2+4+7+14 = 28. Or using the other method, (1+2+4+7+14+28) divided by 2, it results as 28, thus making 28 a perfect number.

Throughout history, there have been studies on perfect numbers. It is not clarified when perfect numbers were first studied and it’s possible that the first studies may go back to the earliest era when numbers first aroused curiosity. It is assumed, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked. Perfect numbers were studied by Pythagoras and his followers, mainly for their mystical properties than for their number theoretic properties. Although, the four perfect numbers 6, 28, 496 and 8128 seem to have been known from ancient times and there is no record of these discoveries. The first recorded mathematical result concerning perfect numbers which is known occurs in Euclid's Elements written around 300BC. Also, it’s quite interesting to know that, not all perfect numbers were found in the same order, the ninth, tenth, and eleventh perfect numbers were found after the twelfth was discovered. The 29th was found...

...Semester, 2009
History of Philosophy
PLTL 1111 AA
THE DIVINITY OF NUMBER:
The Importance of Number in the Philosophy of Pythagoras
by
Br. Paul Phuoc Trong Chu, SDB
Pythagoras and his followers, the Pythagoreans, were profoundly fascinated with numbers. In this paper, I will show that the heart of Pythagoras’ philosophy centers on numbers. As true to the spirit of Pythagoras, I will demonstrate this in seven ways. One, the principle of reality is mathematics and its essence is numbers. Two, odd and even numbers signify the finite and infinite. Three, perfectnumbers correspond with virtues. Four, the generation of numbers leads to an understanding of the One, the Divinity. Five, the tetractys is important for understanding reality. Six, the ratio of numbers in the tetractys governs musical harmony. Seven, the laws of harmony explain workings of the material world.
The Pythagoreans “believed that [the principles of mathematics] are the principles of all things that are”. Further, “number is the first of these principles”.[1] “’The numerals of Pythagoras,’ says Porphyry, who lived about 300 A. D., ‘were hieroglyphic symbols, by means whereof he explained all ideas concerning the nature of things…’”[2] In modern time, we can see clearly the application of mathematical principles in our daily...

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Now into its eighth edition and with additional material on primality testing, written by J. H. Davenport, The Higher Arithmetic introduces concepts and theorems in a way that does not require the reader to have an in-depth knowledge of the theory of numbers but also touches upon matters of deep mathematical signiﬁcance. A companion website (www.cambridge.org/davenport) provides more details of the latest advances and sample code for important algorithms. Reviews of earlier editions: ‘. . . the well-known and charming introduction to number theory . . . can be recommended both for independent study and as a reference text for a general mathematical audience.’ European Maths Society Journal ‘Although this book is not written as a textbook but rather as a work for the general reader, it could certainly be used as a textbook for an undergraduate course in number theory and, in the reviewer’s opinion, is far superior for this purpose to any other book in English.’ Bulletin of the American Mathematical Society
THE HIGHER ARITHMETIC
AN INTRODUCTION TO THE THEORY OF NUMBERS
Eighth edition
H. Davenport
M.A., SC.D., F.R.S.
late Rouse Ball Professor of Mathematics in the University of Cambridge and Fellow of Trinity College Editing and additional material by
James H. Davenport
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town,...

...What's Special About This Number?
0 is the additive identity.
1 is the multiplicative identity.
2 is the only even prime.
3 is the number of spatial dimensions we live in.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the number of Platonic solids.
6 is the smallest perfectnumber.
7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass.
8 is the largest cube in the Fibonacci sequence.
9 is the maximum number of cubes that are needed to sum to any positive integer.
10 is the base of our number system.
11 is the largest known multiplicative persistence.
12 is the smallest abundant number.
13 is the number of Archimedian solids.
14 is the smallest even number n with no solutions to φ(m) = n.
15 is the smallest composite number n with the property that there is only one group of order n.
16 is the only number of the form xy = yx with x and y being different integers.
17 is the number of wallpaper groups.
18 is the only positive number that is twice the sum of its digits.
19 is the maximum number of 4th powers needed to sum to any number.
20 is the number of rooted trees with 6 vertices.
21 is the smallest...

...for positive integer m,
[pic]
where [pic]means "evenly divides". Substituting a = 2r, b = − 1, and m = s and using that s is odd,
[pic]
and thus
[pic]
Because 1 < 2r + 1 < 2n + 1, it follows that 2n + 1 is not prime. Therefore, by contraposition n must be a power of 2.
Well my thought is suppose n isn't a power of 2.
Suppose it's 3.
a^3 + 1
No, that's not prime. if n is odd and n>=3, then I can factorize like this:
(a + 1) (a^2 - a + 1)
and both numbers are greater than one, so a^3+1 can't be prime.
What if n is 6?
a^6 + 1
Well now I can just write that as (a^2)^3 + 1
which we know will factorize the same way:
((a^2) + 1) ( (a^2)^2 - (a^2) + 1)
The same idea will work for any number with an odd factor of 3 or more.
Therefore the only numbers n that can possibly lead to primes are those that have no odd factors greater than 1.
In other words, they are exactly the powers of 2.
Read more: If a^n+1 is prime for some number a>=2 and n>=1, show that n must be a power of 2 | Answerbag http://www.answerbag.com/q_view/1122154#ixzz1k6wypEZ9
>(b) if a^n + 1 is prime, show that a is even and that n is a power
> of 2.
If n is not a power of two, then it has an odd prime factor m.
So we can write n = m*r where we know for sure that m is odd. Since m
is odd, we can write m = 2*k + 1 for some integer k.
Thus n = 2*k*r + r
Now exhibit a...

...REAL NUMBERS
Q.1 Determine the prime factorization of the number 556920. (1 Mark)
(Ans) 23 x 32 x 5 x 7 x 13 x 17
Explanation :
Using the Prime factorization, we have
556920 = 2 x 2 x 2 x 3 x 3 x 5 x 7 x 13 x 17 = 23 x 32 x 5 x 7 x 13 x 17
Q.2 Use Euclid’s division algorithm to find the HCF of 210 and 55. (1 Mark)
(Ans) 5
Explanation:
5 , Given integers are 210 and 55 such that 210 > 55. Applying Euclid’s division leema to 210 and 55, we get
210 = 55 x 3 + 45 ……….(1)
55 = 45 x 1 +10 ………(2)
45 = 10 x 4 + 5 ………..(3)
10 = 5 x 2 + 0 ………..(4)
we consider the new divisor 10 and the new remainder 5 and apply division leema to get 10 = 5 x 2 + 0 The remainder at this stage is zero. So, the divisor at this stage or the remainder at the previous stage i.e.5 is the HCF of 210 and 55.
Q.3 The areas of three fields are 165m2 , 195m2 and 285m2respectively. From these flowers beds of equal size are to be made. If the breadth of each bed be 3 metres, what will be the maximum length of each bed? (1 Mark)
(Ans) 4m
Explanation :
The area of three fields are 165 m2, 195 m2and 285 m2. Maximum area of a flower bed = HCF of 165, 195 and 285We first find the HCF of 165 and 195. Using Euclid's algorithm, we have the following equations.
195 = 165 × 1 + 30
165 = 30 × 5+ 15
30 = 15 × 2 + 0
The remainder has now become zero, so our procedure stops.
Since the divisor at this stage is 15.
HCF (165, 195) = 15
Now we...

...10th Real Numbers test paper
2011
1.
Express 140 as a product of its prime factors
2.
Find the LCM and HCF of 12, 15 and 21 by the prime factorization method.
3.
Find the LCM and HCF of 6 and 20 by the prime factorization method.
4.
State whether13/3125 will have a terminating decimal expansion or a non-terminating repeating
decimal.
5.
State whether 17/8 will have a terminating decimal expansion or a non-terminating repeating
decimal.
6.
Find the LCM and HCF of 26 and 91 and verify that LCM × HCF = product of the two numbers.
7.
Use Euclid’s division algorithm to find the HCF of 135 and 225
8.
Use Euclid’s division lemma to show that the square of any positive integer is either of the form
3m or 3m + 1 for some integer m
9.
Prove that √3 is irrational.
10. Show that 5 – √3 is irrational
11. Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some
integer.
12. An army contingent of 616 members is to march behind an army band of 32 members in a parade.
The two groups are to march in the same number of columns. What is the maximum number of
columns in which they can march?
13. Express 156 as a product of its prime factors.
14. Find the LCM and HCF of 17, 23 and 29 by the prime factorization method.
15. Find the HCF and LCM of 12, 36 and 160, using the prime factorization method.
16. State whether 6/15 will have...

...he number theory or number systems happens to be the back bone for CAT preparation. Number systems not only form the basis of most calculations and other systems in mathematics, but also it forms a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems there are hundreds of concepts and variations, along with various logics attached to them, which makes this seemingly easy looking topic most complex in preparation for the CAT examination. The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on this section.
Real numbers: The numbers that can represent physical quantities in a complete manner. All real numbers can be measured and can be represented on a number line. They are of two types:
Rational numbers: A number that can be represented in the form p/q where p and q are integers and q is not zero. Example: 2/3, 1/10, 8/3 etc. They can be finite decimal numbers, whole numbers, integers, fractions.
Irrational numbers: A number that cannot be...

... 3 is a number, numeral, and glyph. It is the natural number following 2 and preceding 4.
In mathematics
Three is approximately π when doing rapid engineering guesses or estimates. The same is true if one wants a rough-and-ready estimate of e, which is actually approximately 2.71828.
Three is the first odd prime number, and the second smallest prime. It is both the first Fermat prime and the first Mersenne prime, the onlynumber that is both, as well as the first lucky prime. However, it is the second Sophie Germain prime, the second Mersenne prime exponent, the second factorial prime, the second Lucas prime, the second Stern prime.
Three is the first unique prime due to the properties of its reciprocal.
Three is the aliquot sum of 4.
Three is the third Heegner number.
According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.
Three is the second triangular number and it is the only prime triangular number. Three is the only prime which is one less than a perfect square. Any other number which is n2 − 1 for some integer n is not prime, since it is . This is true for 3 as...