Bulletin of the Marathwada Mathematical Society
Vol. 10, No. 2, December 2009, Pages 28–34.
DIFFERENT TYPES OF NATURAL NUMBERS
G. S. Kawale,
Department of Mathematics
Vasantrao Naik Mahavidyalaya,
Aurangabad,(M. S.), India.
Abstract
Some typical numbers like amicable,polite, lucky, cyclic, rare,rotating etc are
defined and illustrated with the help of examples. The interesting properties of
such numbers are also discussed.
1
INTRODUCTION
The commonly recognized five kinds of numbers are Natural or Positive Integers, Integers, Rational, Real and Complex.The natural numbers of different types
like Triangular, Square, Pentagonal etc are known.
However, in addition to these, following types of natural numbers are of interest:
(i) Palindromic numbers (ii) Amicable numbers (iii) Rotating numbers (iv)
Lucky numbers (v) Polite numbers (vi) Cyclic numbers (vii) Rare numbers.
Algorithms can be prepared for producing these numbers and are given for
some typical numbers. A brief History of these numbers is also given wherever
possible.
2
2.1
TYPES OF NATURAL NUMBERS
Palindromic Numbers
The name palindrom arises from the English word palindrom. The numbers
which when read from left and right are the same are known as palindromic numbers.
Some Palindromic numbers can be obtained by using the following method:
(i) Choose any number. (ii) Reverse its digits. (iii) Add the number and its
reverse number.
If palindromic number is not formed, repeat the process till a palindromic
28
DIFFERENT TYPES OF NATURAL NUMBERS
29
number is obtained.
Example
87 + 78 = 165
165 + 561 = 726
726 + 627 = 1353
1353 + 3531 = 4884.
4884 is a palindromic number.
196 is the only number not giving palindromic number in 10,000 steps. The
mathematician Leiland started from 196 and repeated the above process 50,000
times but he could not reach a palindromic number.The mathematician Anderton
repeated the process 70,928 times starting from 196 but a palindromic number could
not be obtained.
In one lac steps , the number 5996 does not give a palindromic number.
ˆ Number 22 and its square 484 are both palindromic. Similarly some powers
of 11 are palindromic.
ˆ The sum and product of two palindromic numbers need not be always a palindromic number.
ˆ Some palindromic numbers are prime numbers. Foe example, 101, 131, 151,
181, 313, 353, 727, 757, 797, 919, 79997, 91019, 93139, 93739, 94049 etc are
prime and palindromic.
2.2
Amicable Numbers
Two integers x and y are said to be amicable if each of them can be expressed
as the sum of all the divisors of the other, except that number. Foe example 284
and 220 are amicable numbers because the sum of the divisors of 220 is
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284
and the sum of the divisors of 284 is
1 + 2 + 4 + 71 + 142 = 220.
A procedure for finding out amicable numbers is as follows:
If a = (3)(2x ) − 1, b = (3)(2x−1 ) − 1, c = 9(22x−1 ) − 1, (x > 1) and if a, b, c are
prime numbers, then 2x ab and 2x c are amicable numbers. For example if x = 4,
then 2x ab = 24 × 47 × 23 = 17296, and 2x c = 24 × 1151 = 18416. It can be verified
that 17296 and 18416 are amicable.
The invention of amicable numbers was done by ancient Greek people.
Pythagoras was familiar with amicable numbers 220 and 284. In 1636 Fermat found
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G. S. Kawale
second pair of amicable numbers i. e. 17296 and 18416. Descarte in 1638 found
the third pair of amicable numbers i. e. 9363584 and 9437056. Majority of pairs
of amicable numbers were discovered by Euler. He obtained 60 pairs of amicable
numbers.
ˆ Amicable numbers are either both even or both odd.
ˆ Odd amicable numbers are divisible by 3 and even amicable numbers are
divisible by 9.
The reason for the non-existence of an odd and an even number in a pair of
amicable numbers is not known.
Some pairs of Amicable numbers are given in the following chart.
Sr. No.
1
2
3
4
5
6
2.3
First No.
220
1184
2620
5020
10744
17296
Second No.
284
1210
2924
5564
10856
18416
Rotating Numbers
A number x is said to be rotating if x can be obtained by following a definite
procedure described below.
(a) Divide the number x in two parts.
(b) Take squares of two parts separately and add.
(c) Repeat the steps (a) and (b) till the original number x is obtained.
For example start with 4756.
4756 :
5345 :
4834 :
3460 :
472 + 562
532 + 452
482 + 342
342 + 602
= 5345,
= 4834,
= 3460,
= 4756,
Hence 4756 is a rotating number.We required four steps.
An example of a rotating number requiring five steps is given below.
DIFFERENT TYPES OF NATURAL NUMBERS
3425 −→ 342 + 252
1781 −→ 172 + 812
6850 −→ 682 + 502
7124 −→ 712 + 242
5617 −→ 562 + 172
31
= 1781,
= 6850,
= 7124,
= 5617,
= 3425,
Hence 3425 is a rotating number.
1233 is a rotating number obtained in only one step.
It can be verified that no two digit number is rotating except 10.
2.4
Polite and Impolite Numbers
Numbers which can be expressed as the sum of consecutive numbers are called
Polite numbers.
For example the numbers 5 = 2 + 3, 22 = 4 + 5 + 6 + 7, 19 = 9 + 10 are all
polite.It can be noted that every odd number > 1 is polite.
The numbers which are not expressible in the form of sum of consecutive numbers are called Impolite numbers. For example 1, 2, 4, 8, 16, 32 etc are not polite
numbers.
Impolite numbers can be written in the form of 2n where
n = 1, 2, 3, · · ·
Formula for obtaining Polite numbers is
N=
n(2a + n − 1)
2
where a and n are integers, a ≥ 1 and n ≥ 2.
Actually this is a formula for a sum of n consecutive numbers in Arithmetic Progression, with the first term = a. If n is even number, then (2a + n − 1) is odd and
if n is odd, then (2a + n − 1) is even number.
2.5
Cyclic Numbers
The number 142857 is amazing, since upon multiplication by 1 to 6 the following numbers are obtained, having the same digits.
142857 × 1 = 142857,
142857 × 2 = 285714,
142857 × 3 = 428571,
142857 × 4 = 571428,
142857 × 5 = 714285,
142857 × 6 = 857142.
Thus, the numbers obtained by multiplying 142857 by 1 to 6 produce numbers
which contain 1, 4, 2, 8, 5 and 7 in different order. Such numbers are called as cyclic
numbers.
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G. S. Kawale
The number 147857 is obtained by writting 1/7 in the decimal form.
1/7 = 0.142857142857142857 · · · = 0.142857.
Interestingly, if 1 is divided by 7, the repetition in same sequence of remainders i.e.
1,3,2,6,4 and 5 is observed. so the number 142857 becomes a cyclic number.
Other interesting property of the cyclic number 142857 is that upon multiplication
by 7 it gives 999999.
The cyclic numbers can be obtained by considering the decimal expansion of
etc. For example
1
1
1
1
1
13 , 17 , 19 , 23 , 29 ,
1
= 0.052631578947368421 · · ·
19
In this representation there are 18 digits after decimal point and these are repeated in
the same order after 18th digit. The number 052631578947368421 is a cyclic number.
It is interesting to observe the digits in the numbers obtained by multiplying this
number by 2, 3, 4, · · · 18.
2.6
Lucky Numbers
To find out Lucky Numbers following method may be adopted.
Step 1: Write natural numbers in order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, · · ·
Step 2 : Delete all even numbers, giving, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, · · ·
In step 2, number 3 is present after 1. Now delete every third number from
step 2, that is drop out 5, 11, 17, · · · The remaining numbers are,
1, 3, 7, 9, 13, 15, 19, 21, 25, 27, 31, 33,
After 3, number 7 is present.
Now drop out every seventh number from this list. Completion of this process gives
Lucky numbers. The list of Lucky numbers is as follows:
1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, · · ·
2.7
Perfect Numbers
A number x is called a perfect number if it is equal to the sum of all its divisors
except x.
DIFFERENT TYPES OF NATURAL NUMBERS
33
It is easy to show that numbers 6 and 28 are perfect numbers, because
6 = 1 + 2 + 3,
28 = 1 + 2 + 4 + 7 + 14.
The formula for perfect numbers as given by Euclid is N = (2n−1 )(2n − 1)
provided the value of n is such that (2n − 1) is a prime number. Note that every
number of the form (2n − 1) is not a prime. For example if n = 4, then we get 15,
which is not a prime number, and hence the number (24−1 )(24 − 1) i.e. 120 is not a
perfect number.
The first six perfect numbers are 6, 28, 496, 8128, 33550336 and 8589869056.It
is easy to observe that a perfect number cannot be odd.
Now we shall show that the number P (n) given by P (n) = (2n−1 )(2n − 1) is
perfect, whenever (2n − 1) is prime.
Theorem 2.1 If (2n −1) is a prime number, then (2n−1 )(2n −1) is a perfect number.
Proof: For a positive integer n, we define σ(n) as the sum of all positive divisors
of n, including 1 and n. By definition of a perfect number, it follows that if N is a
perfect number, then σ(N ) = 2N.
Now let N = (2n−1 )(2n − 1) then by putting (2n − 1) = p, a prime number,
we can write
N = 2n−1 · p.
Since p is prime, the divisors of N are 1, 2, 22 , 23 , · · · , 2n−1 , p, 2p, · · · , 2n−1 · p.
Hence
σ(N ) = 1 + 2 + 22 + 23 + · · · + 2n−1 + p + 2p + · · · + 2n−1 · p
= (2n − 1) + p(2n − 1),
= (2n − 1)(p + 1) = (2n − 1)(2n − 1 + 1),
= 2n (2n − 1) = 2[2n−1 (2n − 1)] = 2N.
Hence N is a perfect number.
2.8
Rare Numbers
If sum and difference of a natural number x and the number obtained by reversing its digits give numbers which are both perfect squares then x is called a rare
number.Rare numbers are found rarely.
In first 1000 crore natural numbers there are only five numbers which are rare.
These are 65, 621770, 281089082, 2922652202 and 2042832002.
Illustration :
65 − 56 = 32 , 65 + 56 = 112
621770 − 77126 = 7382 , 621770 + 77126 = 8362 . etc
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G. S. Kawale
Acknowledgement: The guidance given by Dr. S. R. Joshi for the preparation of
this paper is gratefully acknowledged.
References
[1] Apte Mohan,
Edition, 2001.
.
Ashwamedh Prakashan, Pune, Second