Post by 1dave on Jun 2, 2023 14:13:48 GMT -5
sites.google.com/site/mathematicsmiscellany/very-special-numbers
The classification of numbers into natural numbers, negative numbers, rational numbers, irrational numbers, complex numbers and transcendental numbers is common knowledge for students and lovers of mathematics. It has been established in modern algebra that the field of complex numbers is sufficient for accommodating the roots of all polynomials.
We now consider a few numbers with which human qualities and social attributes have been attached.
PERFECT NUMBER: Suppose n is a natural number. Let σ(n) represent the sum of all the natural divisors of n, including 1 but excluding itself.
Three cases can arise: (i) σ(n) = n, (ii) σ(n) > n, and (iii) σ(n) < n.
Case (i) σ(n) = n. Such numbers are called perfect numbers. Some examples of this type of numbers are given below:
n = 6 = 1 + 2 + 3 = σ(n)
n = 28 = 1 + 2 + 4 + 7 + 14 = 28 = σ(n)
n = 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = σ(n)
n = 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = σ(n) = 26(27 – 1)
The number 8128 is also called a harmonic divisor number or Ore number, which is defined as follows:
If d (n) is the number of positive integer divisors of n, then if (n d(n))/σ(n) is an integer, then n is called a harmonic divisor number or Ore number.
To consider an example: let n = 140; σ(n) = 1 + 2 + 4 + 5 + 7 + 14 + 20 + 28 + 35 + 70 + 140 = 336. d(n) or (the number of positive divisors of n) = 12.
(n d(n))/ σ(n) = (140.12)/336 = 5, which is an integer. Thus, 140 is a harmonic or Ore number.
When n = 8128, d(n) = 13. Now, (8128 x 13)/8128 = 13, an integer. Hence, 8128 is a harmonic number or harmonic divisor number or Ore Number.
ABUNDANT NUMBERS: An abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n.
The first few abundant numbers are:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, …
As an example, consider the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − (2 × 24) = 12.
The smallest odd abundant number is 945. The smallest abundant number, not divisible by 2 or by 3 is 5391411025 whose prime factors are 5, 7, 11, 13, 17, 19, 23 and 29.
DEFICIENT/DEFECTIVE NUMBERS: A deficient number or defective number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the numbers deficiency.
Examples:
The first few deficient numbers are:
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, …
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 2 × 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
CONGRUENT NUMBERS: Two numbers a and b which leave the same reminder when divided by a certain number n, called the modulus, are called congruent numbers and this is expressed in the form:
a ≡ b (mod n).
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of integer congruent numbers starts with
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, …
For example, 5 is a congruent number because it is the area of a 20/3, 3/2, 41/6 triangle. Similarly, 6 is a congruent number because it is the area of a 3,4,5 triangle. 7 is a congruent number because it is the area of a 35/12, 24/5, 337/60 triangle (a congruent number is an integer that is the area of a right triangle with three rational number sides). 3 is not a congruent number.
FRIENDLY NUMBERS/SOLITARY NUMBERS: In number theory, a friendly number is a natural number that shares a certain characteristic called abundancy, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers with the same abundancy form a friendly pair.
Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually friendly numbers.
Some of the friendly pairs are: 6 and 28, 30 and 140, 80 and 200, 210 and 224, 12 and 234, 66 and 308, 66 and 308, 84 and 270.
A number that is not part of any friendly pair is called solitary.
The abundancy of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Note that abundancy is not the same as abundance which is defined as σ(n) − 2n..
The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming for example the friendly pair (6, 28) with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases.
One of the more famous unsolved problems in mathematics is the question of whether 10 is a solitary number.
Numbers which have no friendly pair are called 'solitary' numbers. There are not many friendly numbers--true friends are rare indeed. So most of the numbers are solitary! And it is pretty obvious that it may be a bit hard to prove for some big number whether it has a friendly pair or not, but it is difficult to prove that 10 has no friendly pair. This remains an unsolved problem.
So far people have proved for some numbers that they are NOT solitary by finding their friendly pair. Sometimes the difference in pairs is very large: for example, 24 has a friendly pair in 91963648. But no such pair has ever been found for 10.
4320, 4680 form a friendly pair; σ(4320) ≡ 15120, and σ(4680) ≡ 16380, and 15120/4320 = 7/2, and 16380/4680 = 7/2.
Another example is 24 and 91963648, for which the ratio happens to be 5/2.
CARMICHAEL NUMBER: A Carmichael number is an odd composite number n which satisfies Fermat's little theorem
an-1 – 1 ≡ 0 mod n
for every choice of a satisfying (a,n) = 1 (i.e., a and n are relatively prime) with 1 < a < n.
The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ... The number of Carmichael numbers less than 102 and 103, ... are 0, 1, 7, 16, 43, 105, .... The smallest Carmichael numbers having 3, 4, ... factors are 561 = 3 x 11 x 17, 41041 = 7 x 11 x 13 x41, 825265, 321197185, ...
HARSHAD/NIVEN NUMBER: A number that is divisible by the sum of its own digits. For example, 1729 is a Harshad number because 1 + 7 + 2 + 9 = 19 and 1729 = 19 × 91.
More generally, a Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base.
2620 and 2924 are Harshad amicable pairs, since 2620 is divisible by (2 + 6 + 2 + 0) or 10, and 2924 is divisible by (2 + 9 + 2 + 4) or 17, since 2924 = 17 x 172.
AMICABLE NUMBERS: Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers is equal to the other. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3).
Two numbers are called Amicable (or friendly) if each equals to the sum of the aliquot divisors of the other (aliquot divisors means all the divisors excluding the number itself). For example aliquot divisors of number 220 are 1,2,4,5,10,11,20,22,44,55 and 110. The aliquot divisors of number 284 are 1,2,4,71 and 142.
If we represent an amicable pair (AP) by (m, n,) and sum of aliquot divisors of m and n by s (m) and s (n) respectively, then (220, 284) form an amicable pair since σ(m) = σ (220) = 1+2+4+5+10+11+20+22+44+55+110 = 284 = n, and σ (n) = σ (284) = 1+2+4+71+142 = 220 = m
In 1636 Pierre de Fermat discovered another pair of amicable numbers (17296, 18416). Later Descartes gave the third pair of amicable numbers i.e. (9363584, 9437056). These results were actually rediscoveries of numbers known to Arab mathematicians. In the 18th century great Euler drew up a list of 64 amicable pairs. B.N.I. Paganini, a 16 years Old Italian, startled the mathematical world in 1866 by announcing that the numbers 1184 and 1210 were friendly. It was the second lowest pair and had been completely overlooked until then. Till 28 Sept, 2007 about 11994387 pairs of amicable numbers are known.
(17296, 18416) also form an amicable pair.
SIERPINSKI NUMBER: In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that k2n + 1 is composite, for all natural numbers n; in 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.
In other words, when k is a Sierpinski number, all members of the following set are composite:
{k.2n + 1: n is a Natural number)
Numbers in this set with odd k and k < 2n are called Proth numbers.
The sequence of currently known Sierpinski numbers begins with:
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, …
The number 78,557 was proved to be a Sierpinski number by John Selfridge in 1962. Another known Sierpinski number is 271129.
RIESEL NUMBER: In mathematics, a Riesel number is an odd natural number k for which the integers of the form k·2n − 1 are composite for all natural numbers n.
In other words, when k is a Riesel number, all members of the following set are composite:
{k.2n – 1: n is a natural number)
In 1956, Hans Riesel showed that there are an infinite number of integers k such that k·2n − 1 is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.
A number can be shown to be a Riesel number by exhibiting a covering set: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
509203×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
762701×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
777149×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
790841×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
992077×2n − 1 has covering set {3, 5, 7, 13, 17, 241}.
The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, sixty-one values of k less than this have yielded only composite numbers for all values of n so far tested. The smallest of these are 2293, 9221, 23669, 31859, 38473, 40597, 46663, 65531, 67117 and 74699.
SOCIABLE NUMBER: Sociable numbers are numbers that result in a periodic aliquot sequence, where an aliquot sequence is the sequence of numbers obtained by repeatedly applying the restricted divisor function to n and σ(n) is the usual divisor function.
If the period of the aliquot cycle is 1, the number is called a perfect number. If the period is 2, the two numbers are called an amicable pair. In general, if the period is t ≥ 3, the number is called sociable of order t. For example, 1264460 is a sociable number of order four since its aliquot sequence is 1264460, 1547860, 1727636, 1305184, 1264460, ....
Only two groups of sociable numbers were known prior to 1970, namely the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4.
The first few sociable numbers are 12496, 14316, 1264460, 2115324, 2784580, 4938136, ... which have orders 5, 28, 4, 4, 4, 4, .... Excluding perfect numbers, a total of 152 sociable cycles are known as of Feb. 2009.
KAPREKAR NUMBER: Consider an n-digit number k. Square it and add the right n digits to the left n or n-1 digits. If the resultant sum is k, then k is called a Kaprekar number. For example, 9 is a Kaprekar number since
92 = 81, and 8 + 1 = 9
and 297 is a Kaprekar number since
2972 = 88209 and 88 + 209 = 297.
452 = 2025 and 20 + 25 = 45.
552 = 3025, and 30 + 25 = 55
7032 = 494209 and 494 + 205 = 703.
27282 = 7441984 and 744 + 1984 = 2728
52922 = 28005264 and 28 + 005264 = 5292
8571432 = 734294122449 and 734694 + 122449 = 857143.
The first few are 1, 9, 45, 55, 99, 297, 703, 999,….
However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.
KAPREKAR CONSTANT: Kaprekar discovered the Kaprekar constant or 6174 in 1949. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have
4321 − 1234 = 3087, then
8730 − 0378 = 8352, and
8532 − 2358 = 6174.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.
However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for more digits (or 2), the numbers enter into one of several cycles.
Kaprekar's Constant for 3-Digit Numbers: 495
The Kaprekar transformation for three digits involving the number 495 is defined as follows:
1) Take any three-digit number with at least two digits different.
2) Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
3) Subtract the smaller number from the bigger number.
4) Go back to Step #2.
The above operation will always reach 495 in a few steps where it stops there.
Once 495 is reached, the process will keep yielding 954 – 459 = 495.
The only three-digit numbers for which this does not hold are numbers whose digits are all identical (e.g., 222) as they yield zero after a single iteration.
Example #1:
100 yields:
100–001=099
990–099=891
981–189=792
972–279=693
963-369=594
954 – 459 = 495
Example #2:
912 yields:
921–129=792
972–279=693
963–369=594
954 – 459 = 495
Kaprekar constant for 4-digit numbers is 6174.
You can take any four-digit number and re-arrange the digits in decreasing order. All digits MUST be different. We'll use 4521 - let's order the digits from highest to lowest which gives us 5421.
2. Now take the number and order the digits from lowest to highest and subtract from the number you ordered from high to low.(Repeat the process until you come to the Constant of 6174)
Original number: 4521
5421-1245=4176
7641-1467 = 6174
After going through the process twice, we reach 6174. Try another 4 digit number:
9472
9742-2479=7263
7632-2367=5265
6552-2556=3996
9963-3699=6264
6642-2466=4176
7641-1467= 6174
Series for 2-digit numbers: There is only one series for 2-digit numbers - 9 -> 81 -> 63 -> 27 -> 45 -> repeat
Series for 5-digit numbers
There are three series for 5-digit numbers -
74943 -> 62964 -> 71973 -> 83952 -> repeat
63954 -> 61974 -> 82962 -> 75933 -> repeat
53955 -> 59994 -> repeat
Series for 6-digit numbers
There are two Kaprekar Constants for 6-digit numbers - 631764 and 549945. And there is one series -
851742 -> 750843 -> 840852 -> 860832 -> 862632 -> 642654 -> 420876 -> repeat
Series for 7-digit numbers
There is only one series for 7-digit numbers -
8429652 -> 7619733 -> 8439552 -> 7509843 -> 9529641 -> 8719722 -> 8649432 -> 7519743 -> repeat
Series for 8-digit numbers
There are two Kaprekar Constants for 8-digit numbers - 97508421 and 63317664. And there are two series -
86526432 -> 64308654 -> 83208762 -> repeat
86308632 -> 86326632 -> 64326654 -> 43208766 -> 85317642 -> 75308643 -> 84308652 -> repeat
Series for 9-digit numbers: There are two Kaprekar Constants for 9-digit numbers - 864197532 and 554999445. And there is one series -
865296432 -> 763197633 -> 844296552 -> 762098733 -> 964395531 -> 863098632 -> 965296431 -> 873197622 -> 865395432 -> 753098643 -> 954197541 -> 883098612 -> 976494321 -> 874197522 -> repeat
Series for 10-digit numbers: There are three Kaprekar Constants for 10-digit numbers - 9753086421, 6333176664 and 9975084201. And there are five series -
8655264432 -> 6431088654 -> 8732087622 -> repeat
8653266432 -> 6433086654 -> 8332087662 -> repeat
8765264322 -> 6543086544 -> 8321088762 -> repeat
8633086632 -> 8633266632 -> 6433266654 -> 4332087666 -> 8533176642 -> 7533086643 -> 8433086652 -> repeat
9775084221 -> 9755084421 -> 9751088421 -> repeat
FERMAT NUMBER: The n -th Fermat number is defined as:
Fn = 22n + 1.
Fermat incorrectly conjectured that all these numbers were primes, although he had no proof. The first 5 Fermat numbers: 3 5 17 257 65537 (corresponding to n=0 1 2 3 4 ) are all primes (so called Fermat primes) Euler was the first to point out the falsity of Fermat's conjecture by proving that 641 is a divisor of F5 . (In fact, F5 = 641 6700417 ). Moreover, no other Fermat number is known to be prime for n 4 , so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not.
MERSENNE NUMBERS: In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two:
Some definitions of Mersenne numbers require that the exponent p be prime, since the associated number must be composite if p is.
The largest prime number (243,112,609 − 1) is a Mersenne prime.
Wieferich prime: It is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. The only known Wieferich primes are 1093 and 3511
The classification of numbers into natural numbers, negative numbers, rational numbers, irrational numbers, complex numbers and transcendental numbers is common knowledge for students and lovers of mathematics. It has been established in modern algebra that the field of complex numbers is sufficient for accommodating the roots of all polynomials.
We now consider a few numbers with which human qualities and social attributes have been attached.
PERFECT NUMBER: Suppose n is a natural number. Let σ(n) represent the sum of all the natural divisors of n, including 1 but excluding itself.
Three cases can arise: (i) σ(n) = n, (ii) σ(n) > n, and (iii) σ(n) < n.
Case (i) σ(n) = n. Such numbers are called perfect numbers. Some examples of this type of numbers are given below:
n = 6 = 1 + 2 + 3 = σ(n)
n = 28 = 1 + 2 + 4 + 7 + 14 = 28 = σ(n)
n = 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = σ(n)
n = 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = σ(n) = 26(27 – 1)
The number 8128 is also called a harmonic divisor number or Ore number, which is defined as follows:
If d (n) is the number of positive integer divisors of n, then if (n d(n))/σ(n) is an integer, then n is called a harmonic divisor number or Ore number.
To consider an example: let n = 140; σ(n) = 1 + 2 + 4 + 5 + 7 + 14 + 20 + 28 + 35 + 70 + 140 = 336. d(n) or (the number of positive divisors of n) = 12.
(n d(n))/ σ(n) = (140.12)/336 = 5, which is an integer. Thus, 140 is a harmonic or Ore number.
When n = 8128, d(n) = 13. Now, (8128 x 13)/8128 = 13, an integer. Hence, 8128 is a harmonic number or harmonic divisor number or Ore Number.
ABUNDANT NUMBERS: An abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n.
The first few abundant numbers are:
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, …
As an example, consider the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − (2 × 24) = 12.
The smallest odd abundant number is 945. The smallest abundant number, not divisible by 2 or by 3 is 5391411025 whose prime factors are 5, 7, 11, 13, 17, 19, 23 and 29.
DEFICIENT/DEFECTIVE NUMBERS: A deficient number or defective number is a number n for which the sum of divisors σ(n)<2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n)<n. The value 2n − σ(n) (or n − s(n)) is called the numbers deficiency.
Examples:
The first few deficient numbers are:
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, …
As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 2 × 21, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10.
CONGRUENT NUMBERS: Two numbers a and b which leave the same reminder when divided by a certain number n, called the modulus, are called congruent numbers and this is expressed in the form:
a ≡ b (mod n).
In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of integer congruent numbers starts with
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, …
For example, 5 is a congruent number because it is the area of a 20/3, 3/2, 41/6 triangle. Similarly, 6 is a congruent number because it is the area of a 3,4,5 triangle. 7 is a congruent number because it is the area of a 35/12, 24/5, 337/60 triangle (a congruent number is an integer that is the area of a right triangle with three rational number sides). 3 is not a congruent number.
FRIENDLY NUMBERS/SOLITARY NUMBERS: In number theory, a friendly number is a natural number that shares a certain characteristic called abundancy, the ratio between the sum of divisors of the number and the number itself, with one or more other numbers. Two numbers with the same abundancy form a friendly pair.
Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually friendly numbers.
Some of the friendly pairs are: 6 and 28, 30 and 140, 80 and 200, 210 and 224, 12 and 234, 66 and 308, 66 and 308, 84 and 270.
A number that is not part of any friendly pair is called solitary.
The abundancy of n is the rational number σ(n) / n, in which σ denotes the sum of divisors function. A number n is a friendly number if there exists m ≠ n such that σ(m) / m = σ(n) / n. Note that abundancy is not the same as abundance which is defined as σ(n) − 2n..
The numbers 1 through 5 are all solitary. The smallest friendly number is 6, forming for example the friendly pair (6, 28) with abundancy σ(6) / 6 = (1+2+3+6) / 6 = 2, the same as σ(28) / 28 = (1+2+4+7+14+28) / 28 = 2. The shared value 2 is an integer in this case but not in many other cases.
One of the more famous unsolved problems in mathematics is the question of whether 10 is a solitary number.
Numbers which have no friendly pair are called 'solitary' numbers. There are not many friendly numbers--true friends are rare indeed. So most of the numbers are solitary! And it is pretty obvious that it may be a bit hard to prove for some big number whether it has a friendly pair or not, but it is difficult to prove that 10 has no friendly pair. This remains an unsolved problem.
So far people have proved for some numbers that they are NOT solitary by finding their friendly pair. Sometimes the difference in pairs is very large: for example, 24 has a friendly pair in 91963648. But no such pair has ever been found for 10.
4320, 4680 form a friendly pair; σ(4320) ≡ 15120, and σ(4680) ≡ 16380, and 15120/4320 = 7/2, and 16380/4680 = 7/2.
Another example is 24 and 91963648, for which the ratio happens to be 5/2.
CARMICHAEL NUMBER: A Carmichael number is an odd composite number n which satisfies Fermat's little theorem
an-1 – 1 ≡ 0 mod n
for every choice of a satisfying (a,n) = 1 (i.e., a and n are relatively prime) with 1 < a < n.
The first few Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, ... The number of Carmichael numbers less than 102 and 103, ... are 0, 1, 7, 16, 43, 105, .... The smallest Carmichael numbers having 3, 4, ... factors are 561 = 3 x 11 x 17, 41041 = 7 x 11 x 13 x41, 825265, 321197185, ...
HARSHAD/NIVEN NUMBER: A number that is divisible by the sum of its own digits. For example, 1729 is a Harshad number because 1 + 7 + 2 + 9 = 19 and 1729 = 19 × 91.
More generally, a Harshad number, or Niven number in a given number base, is an integer that is divisible by the sum of its digits when written in that base.
2620 and 2924 are Harshad amicable pairs, since 2620 is divisible by (2 + 6 + 2 + 0) or 10, and 2924 is divisible by (2 + 9 + 2 + 4) or 17, since 2924 = 17 x 172.
AMICABLE NUMBERS: Amicable numbers are two different numbers so related that the sum of the proper divisors of one of the numbers is equal to the other. (A proper divisor of a number is a positive integer divisor other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3).
Two numbers are called Amicable (or friendly) if each equals to the sum of the aliquot divisors of the other (aliquot divisors means all the divisors excluding the number itself). For example aliquot divisors of number 220 are 1,2,4,5,10,11,20,22,44,55 and 110. The aliquot divisors of number 284 are 1,2,4,71 and 142.
If we represent an amicable pair (AP) by (m, n,) and sum of aliquot divisors of m and n by s (m) and s (n) respectively, then (220, 284) form an amicable pair since σ(m) = σ (220) = 1+2+4+5+10+11+20+22+44+55+110 = 284 = n, and σ (n) = σ (284) = 1+2+4+71+142 = 220 = m
In 1636 Pierre de Fermat discovered another pair of amicable numbers (17296, 18416). Later Descartes gave the third pair of amicable numbers i.e. (9363584, 9437056). These results were actually rediscoveries of numbers known to Arab mathematicians. In the 18th century great Euler drew up a list of 64 amicable pairs. B.N.I. Paganini, a 16 years Old Italian, startled the mathematical world in 1866 by announcing that the numbers 1184 and 1210 were friendly. It was the second lowest pair and had been completely overlooked until then. Till 28 Sept, 2007 about 11994387 pairs of amicable numbers are known.
(17296, 18416) also form an amicable pair.
SIERPINSKI NUMBER: In number theory, a Sierpinski or Sierpiński number is an odd natural number k such that k2n + 1 is composite, for all natural numbers n; in 1960, Wacław Sierpiński proved that there are infinitely many odd integers k which have this property.
In other words, when k is a Sierpinski number, all members of the following set are composite:
{k.2n + 1: n is a Natural number)
Numbers in this set with odd k and k < 2n are called Proth numbers.
The sequence of currently known Sierpinski numbers begins with:
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, …
The number 78,557 was proved to be a Sierpinski number by John Selfridge in 1962. Another known Sierpinski number is 271129.
RIESEL NUMBER: In mathematics, a Riesel number is an odd natural number k for which the integers of the form k·2n − 1 are composite for all natural numbers n.
In other words, when k is a Riesel number, all members of the following set are composite:
{k.2n – 1: n is a natural number)
In 1956, Hans Riesel showed that there are an infinite number of integers k such that k·2n − 1 is not prime for any integer n. He showed that the number 509203 has this property, as does 509203 plus any positive integer multiple of 11184810.
A number can be shown to be a Riesel number by exhibiting a covering set: a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
509203×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
762701×2n − 1 has covering set {3, 5, 7, 13, 17, 241}
777149×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
790841×2n − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
992077×2n − 1 has covering set {3, 5, 7, 13, 17, 241}.
The Riesel problem consists in determining the smallest Riesel number. Because no covering set has been found for any k less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, sixty-one values of k less than this have yielded only composite numbers for all values of n so far tested. The smallest of these are 2293, 9221, 23669, 31859, 38473, 40597, 46663, 65531, 67117 and 74699.
SOCIABLE NUMBER: Sociable numbers are numbers that result in a periodic aliquot sequence, where an aliquot sequence is the sequence of numbers obtained by repeatedly applying the restricted divisor function to n and σ(n) is the usual divisor function.
If the period of the aliquot cycle is 1, the number is called a perfect number. If the period is 2, the two numbers are called an amicable pair. In general, if the period is t ≥ 3, the number is called sociable of order t. For example, 1264460 is a sociable number of order four since its aliquot sequence is 1264460, 1547860, 1727636, 1305184, 1264460, ....
Only two groups of sociable numbers were known prior to 1970, namely the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4.
The first few sociable numbers are 12496, 14316, 1264460, 2115324, 2784580, 4938136, ... which have orders 5, 28, 4, 4, 4, 4, .... Excluding perfect numbers, a total of 152 sociable cycles are known as of Feb. 2009.
KAPREKAR NUMBER: Consider an n-digit number k. Square it and add the right n digits to the left n or n-1 digits. If the resultant sum is k, then k is called a Kaprekar number. For example, 9 is a Kaprekar number since
92 = 81, and 8 + 1 = 9
and 297 is a Kaprekar number since
2972 = 88209 and 88 + 209 = 297.
452 = 2025 and 20 + 25 = 45.
552 = 3025, and 30 + 25 = 55
7032 = 494209 and 494 + 205 = 703.
27282 = 7441984 and 744 + 1984 = 2728
52922 = 28005264 and 28 + 005264 = 5292
8571432 = 734294122449 and 734694 + 122449 = 857143.
The first few are 1, 9, 45, 55, 99, 297, 703, 999,….
However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 1002=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.
KAPREKAR CONSTANT: Kaprekar discovered the Kaprekar constant or 6174 in 1949. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have
4321 − 1234 = 3087, then
8730 − 0378 = 8352, and
8532 − 2358 = 6174.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.
However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for more digits (or 2), the numbers enter into one of several cycles.
Kaprekar's Constant for 3-Digit Numbers: 495
The Kaprekar transformation for three digits involving the number 495 is defined as follows:
1) Take any three-digit number with at least two digits different.
2) Arrange the digits in ascending and then in descending order to get two four-digit numbers, adding leading zeros if necessary.
3) Subtract the smaller number from the bigger number.
4) Go back to Step #2.
The above operation will always reach 495 in a few steps where it stops there.
Once 495 is reached, the process will keep yielding 954 – 459 = 495.
The only three-digit numbers for which this does not hold are numbers whose digits are all identical (e.g., 222) as they yield zero after a single iteration.
Example #1:
100 yields:
100–001=099
990–099=891
981–189=792
972–279=693
963-369=594
954 – 459 = 495
Example #2:
912 yields:
921–129=792
972–279=693
963–369=594
954 – 459 = 495
Kaprekar constant for 4-digit numbers is 6174.
You can take any four-digit number and re-arrange the digits in decreasing order. All digits MUST be different. We'll use 4521 - let's order the digits from highest to lowest which gives us 5421.
2. Now take the number and order the digits from lowest to highest and subtract from the number you ordered from high to low.(Repeat the process until you come to the Constant of 6174)
Original number: 4521
5421-1245=4176
7641-1467 = 6174
After going through the process twice, we reach 6174. Try another 4 digit number:
9472
9742-2479=7263
7632-2367=5265
6552-2556=3996
9963-3699=6264
6642-2466=4176
7641-1467= 6174
Series for 2-digit numbers: There is only one series for 2-digit numbers - 9 -> 81 -> 63 -> 27 -> 45 -> repeat
Series for 5-digit numbers
There are three series for 5-digit numbers -
74943 -> 62964 -> 71973 -> 83952 -> repeat
63954 -> 61974 -> 82962 -> 75933 -> repeat
53955 -> 59994 -> repeat
Series for 6-digit numbers
There are two Kaprekar Constants for 6-digit numbers - 631764 and 549945. And there is one series -
851742 -> 750843 -> 840852 -> 860832 -> 862632 -> 642654 -> 420876 -> repeat
Series for 7-digit numbers
There is only one series for 7-digit numbers -
8429652 -> 7619733 -> 8439552 -> 7509843 -> 9529641 -> 8719722 -> 8649432 -> 7519743 -> repeat
Series for 8-digit numbers
There are two Kaprekar Constants for 8-digit numbers - 97508421 and 63317664. And there are two series -
86526432 -> 64308654 -> 83208762 -> repeat
86308632 -> 86326632 -> 64326654 -> 43208766 -> 85317642 -> 75308643 -> 84308652 -> repeat
Series for 9-digit numbers: There are two Kaprekar Constants for 9-digit numbers - 864197532 and 554999445. And there is one series -
865296432 -> 763197633 -> 844296552 -> 762098733 -> 964395531 -> 863098632 -> 965296431 -> 873197622 -> 865395432 -> 753098643 -> 954197541 -> 883098612 -> 976494321 -> 874197522 -> repeat
Series for 10-digit numbers: There are three Kaprekar Constants for 10-digit numbers - 9753086421, 6333176664 and 9975084201. And there are five series -
8655264432 -> 6431088654 -> 8732087622 -> repeat
8653266432 -> 6433086654 -> 8332087662 -> repeat
8765264322 -> 6543086544 -> 8321088762 -> repeat
8633086632 -> 8633266632 -> 6433266654 -> 4332087666 -> 8533176642 -> 7533086643 -> 8433086652 -> repeat
9775084221 -> 9755084421 -> 9751088421 -> repeat
FERMAT NUMBER: The n -th Fermat number is defined as:
Fn = 22n + 1.
Fermat incorrectly conjectured that all these numbers were primes, although he had no proof. The first 5 Fermat numbers: 3 5 17 257 65537 (corresponding to n=0 1 2 3 4 ) are all primes (so called Fermat primes) Euler was the first to point out the falsity of Fermat's conjecture by proving that 641 is a divisor of F5 . (In fact, F5 = 641 6700417 ). Moreover, no other Fermat number is known to be prime for n 4 , so now it is conjectured that those are all prime Fermat numbers. It is also unknown whether there are infinitely many composite Fermat numbers or not.
MERSENNE NUMBERS: In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two:
Some definitions of Mersenne numbers require that the exponent p be prime, since the associated number must be composite if p is.
The largest prime number (243,112,609 − 1) is a Mersenne prime.
Wieferich prime: It is a prime number p such that p2 divides 2p − 1 − 1, therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. The only known Wieferich primes are 1093 and 3511