Amicable Pair Amicable Pair An amicable pair (m, n) consists of two integers m, n for which the sum of proper divisors (the divisors excluding the number itself) of one number equals the other. Amicable pairs are occasionally called friendly pairs (Hoffman 1998, p. 45), although this nomenclature is to be discouraged since the numbers more commonly known as friendly pairs are defined by a different, albeit related, criterion. Symbolically, amicable pairs satisfy (1) is the restricted divisor function . Equivalently, an amicable pair (m, n) satisfies (4) is the divisor function . The smallest amicable pair is (220, 284) which has factorizations (5) The quantity (9) in this case, 220 + 284 = 504, is called the pair sum .

The first few amicable pairs are (220, 284), (1184, 1210), (2620, 2924) (5020, 5564), (6232, 6368), (10744, 10856), (12285, 14595), (17296, 18416), (63020, 76084), ... (Sloane's A002025 and A002046 ). An exhaustive tabulation is maintained by D. Moews. In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), although these results were actually rediscoveries of numbers known to Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended to 60.

In 1866, 16-year old B. Nicolò I. Paganini found the small amicable pair (1184, 1210) which had eluded his more illustrious predecessors (Paganini 1866-1867; Dickson 1952, p. 47). There were 390 known amicable pairs as of 1946 (Escott 1946). There are a total of 236 amicable pairs below 108 (Cohen 1970), 1427 below 1010 (te Riele 1986), 3340 less than 1011 (Moews and Moews 1993), 4316 less than (Moews and Moews), and 5001 less than (Moews and Moews). Rules for producing amicable pairs include the Thâbit ibn Kurrah rule rediscovered by Fermat and Descartes and extended by Euler to Euler's rule . A further extension not previously noticed was discovered by Borho (1972).

Pomerance (1981) has proved that (10) for large enough n (Guy 1994). No nonfinite lower bound has been proven. Let an amicable pair be denoted (m, n), and take m < n. (m, n) is called a regular amicable pair of type (i, j) if (11) is the greatest common divisor , (12) M and N are squarefree , then the number of prime factors of M and N are i and j. Pairs which are not regular are called irregular or exotic (te Riele 1986). There are no regular pairs of type for and (13) is even , then (m, n) cannot be an amicable pair (Lee 1969). The minimal and maximal values of found by te Riele (1986) were (14) and (15) te Riele (1986) also found 37 pairs of amicable pairs having the same pair sum .

The first such pair is (609928, 686072) and (643336, 652664), which has the pair sum (16) te Riele (1986) found no amicable n-tuples having the same pair sum for n > 2. However, Moews and Moews found a triple in 1993, and te Riele found a quadruple in 1995. In November 1997, a quintuple and sextuple were discovered.

The sextuple is (1953433861918, 2216492794082), (1968039941816, 2201886714184), (1981957651366, 2187969004634), (1993501042130, 2176425613870), (2046897812505, 2123028843495), (2068113162038, 2101813493962), all having pair sum 4169926656000. Amazingly, the sextuple is smaller than any known quadruple or quintuple, and is likely smaller than any quintuple. The earliest known odd amicable numbers all were divisible by 3.

This led Bratley and McKay (1968) to conjecture that there are no amicable pairs coprime to 6 (Guy 1994, p. 56). However, Battiato and Borho (1988) found a counterexample, and now many amicable pairs are known which are not divisible by 6 (Pedersen). The smallest known example of this kind is the amicable pair (42262694537514864075544955198125, 42405817271188606697466971841875), each number of which has 32 digits. A search was then begun for amicable pairs coprime to 30. The first example was found by Y. Kohmoto in 1997, consisting of a pair of numbers each having 193 digits (Pedersen).

Kohmoto subsequently found two other examples, and te Riele and Pedersen used two of Kohmoto's examples to calculated 243 type- pairs coprime to 30 by means of a method which generates type- pairs from a type- No amicable pairs which are coprime to are currently known. The following table summarizes the largest known amicable pairs discovered in recent years. The largest of these is obtained by defining (17) are all primes, and the numbers (23) are an amicable pair, with each member having decimal digits (Jobling 2005). Jobling 2005 Aliquot Sequence , Amicable Quadruple , Amicable Triple , Augmented Amicable Pair , Breeder , Crowd , Euler's Rule , Friendly Pair , Multiamicable Numbers , Pair Sum , Quasiamicable Pair , Rational Amicable Pair , Sociable Numbers , Super Unitary Amicable Pair , Thâbit ibn Kurrah Rule , Unitary Amicable Pair Alanen, J.; Ore, Ø.; and Stemple, J. "Systematic Computations on Amicable Numbers." Math. Comput.

21, 242-245, 1967. Battiato, S. and Borho, W. "Are there Odd Amicable Numbers not Divisible by Three?" Math. Comput. 50, 633-637, 1988. Borho, W. "On Thabit ibn Kurrah's Formula for Amicable Numbers." Math. Comput. 26, 571-578, 1972. Borho, W. "Some Large Primes and Amicable Numbers." Math. Comput. 36, 303-304, 1981. Borho, W. "Befreundete Zahlen: Ein zweitausend Jahre altes Thema der elementaren Zahlentheorie." In Mathematische Miniaturen 1: Lebendige Zahlen: Fünf Exkursionen. Basel, Switzerland, Birkhäuser, pp. 5-38, 1981. Borho, W. and Hoffmann, H. "Breeding Amicable Numbers in Abundance." Math. Comput.

46, 281-293, 1986. Bratley, P.; Lunnon, F.; and McKay, J. "Amicable Numbers and Their Distribution." Math. Comput. 24, 431-432, 1970. Bratley, P. and McKay, J. "More Amicable Numbers." Math. Comput. 22, 677-678, 1968. Cohen, H. "On Amicable and Sociable Numbers." Math. Comput. 24, 423-429, 1970. Costello, P. "Amicable Pairs of Euler's First Form." J. Rec. Math. 10, 183-189, 1977-1978. Costello, P. "Amicable Pairs of the Form . " Math. Comput. 56, 859-865, 1991. Dubner, H. "New Amicable Pair Record." NMBRTHRY@listserv.nodak.edu posting. Oct.

14, 1997. . Erdos, P. "On Amicable Numbers." Publ. Math. Debrecen 4, 108-111, 1955-1956. Erdos, P. "On Asymptotic Properties of Aliquot Sequences." Math. Comput. 30, 641-645, 1976. Escott, E. B. E. "Amicable Numbers." Scripta Math. 12, 61-72, 1946. García, M. "New Amicable Pairs." Scripta Math. 23, 167-171, 1957. Jobling, P. "A New Largest Known Amicable Pair." . Lee, E. J. "Amicable Numbers and the Bilinear Diophantine Equation." NMBRTHRY@listserv.nodak.edu posting. Mar. 10, 2005. Math. Comput. 22, 181-197, 1968. Lee, E. J.

"On Divisibility of the Sums of Even Amicable Pairs." Math. Comput. 23, 545-548, 1969. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, I." J. Rec. Math. 5, 77-93, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, II." J. Rec. Math. 5, 153-173, 1972. Lee, E. J. and Madachy, J. S. "The History and Discovery of Amicable Numbers, III." J. Rec. Math. 5, 231-249, 1972. Lee, E. J. and Madachy, J. S. Errata to "The History and Discovery of Amicable Numbers, I-III." J. Rec. Math. 6, 53, 164, and 229, 1973. Madachy, J. S.

Madachy's Mathematical Recreations. New York: Dover, pp. 145 and 155-156, 1979. Moews, D. and Moews, P. C. "A Search for Aliquot Cycles and Amicable Pairs." Math. Comput. 61, 935-938, 1993. Moews, D. and Moews, P. C. "A List of Amicable Pairs Below . " Rev. Jan. 8, 1993. . Moews, D. and Moews, P. C. "A List of the First 5001 Amicable Pairs." Rev. Jan. 7, 1996. . Ore, Ø. Number Theory and Its History. New York: Dover, pp. 96-100, 1988. Paganini, B. N. I. Atti della R. Accad. Sc. Torino 2, 362, 1866-1867. Pedersen, J. M. "Known Amicable Pairs." . Pedersen, J. M.

"Various Amicable Pair Lists and Statistics." . Pomerance, C. "On the Distribution of Amicable Numbers." J. reine angew. Math. 293/294, 217-222, 1977. Pomerance, C. "On the Distribution of Amicable Numbers, II." J. reine angew. Math. 325, 182-188, 1981. Root, S. Item 61 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 23, Feb. 1972. #item61 . Sloane, N. J. A. Sequences A002025 /M5414 and A002046 /M5435 in "The On-Line Encyclopedia of Integer Sequences." ~njas/sequences/ . Souissi, M.

Un Texte Manuscrit d'Ibn Al-Banna' Al-Marrakusi sur les Nombres Parfaits, Abondants, Deficients, et Amiables. Karachi, Pakistan: Hamdard Nat. Found., 1975. Speciner, M. Item 62 in Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, p. 24, Feb. 1972. #item62 . te Riele, H. J. J. "Four Large Amicable Pairs." Math. Comput. 28, 309-312, 1974. te Riele, H. J. J. "On Generating New Amicable Pairs from Given Amicable Pairs." Math. Comput. 42, 219-223, 1984. te Riele, H. J. J. "Computation of All the Amicable Pairs Below 1010." Math. Comput.

47, 361-368 and S9-S35, 1986. te Riele, H. J. J.; Borho, W.; Battiato, S.; Hoffmann, H.; and Lee, E. J. "Table of Amicable Pairs Between 1010 and 1052." Centrum voor Wiskunde en Informatica, Note NM-N8603. Amsterdam: Stichting Math. Centrum, 1986. C++ Iteration Statements ( for loop ) | difficultcoding C++ Iteration Statements ( for loop ) C++ Iteration Statements ( for loop ) Iteration Statements (C++) Iteration statements cause statements (or compound statements) to be executed zero or more times, subject to some loop-termination criteria. When these statements are compound statements, they are executed in order, except when either the break statement or the continue statement is encountered. Loops repeat a statement a certain number of times, or while a condition is fulfilled. C++ provides three iteration statements — while, do-while, for.

Each of these iterates until its termination expression evaluates to zero (false), or until loop termination is forced with a break statement. The following table summarizes these statements and their actions; each is discussed in detail in the sections that follow. Iteration Statements Yes Yes The statement part of an iteration statement cannot be a declaration. However, it can be a compound statement containing a declaration. for Statement (C++) Executes statement and loop-expression repeatedly until cond-expression becomes false. For information on the range-based for statement, see Range-based for Statement (C++). for ( init-expression ; cond-expression ; loop-expression ) Use the for statement to construct loops that must execute a specified number of times. The for statement consists of three optional parts, as shown in the following table. for Loop Elements Contents init-expression Before any other element of the for statement. init-expression is executed only once. Control then passes to cond-expression. Often used to initialize loop indices. It can contain expressions or declarations (C++ only). cond-expression Before execution of each iteration of statement, including the first iteration. statement is executed only if cond-expression evaluates to true (nonzero). An expression that evaluates to an integral type or a class type that has an unambiguous conversion to an integral type.

Normally used to test for loop-termination criteria. loop-expression At the end of each iteration of statement. After loop-expression is executed, cond-expression is evaluated. Normally used to increment loop indices. For example: for ( i = 0 ; i < 2 ; i++ ) cout<<i <<endl; The preceding for loop is equivalent to the following while loop: #include <stdio.h> init-expression and loop-expression can contain multiple statements separated by the comma operator. For example: #include <stdio.h> for ( i = 5, j = 10 ; i + j < 20; i++, j++ ) cout<<” i + j = “<< (i + j) ; } A for loop terminates when a break, return, or goto (to a labeled statement outside the for loop) within statement is executed. A continue statement in a for loop terminates only the current iteration. If cond-expression is omitted, it is considered true and the for loop will not terminate without a break, return, or goto within statement. A convenient way to specify an infinite loop using the for statement is: for( ; ; ) // Statements to be executed.

} Although the three fields of the for statement are normally used for initialization, testing for termination, and incrementing, they are not restricted to these uses. For example, the following code prints the numbers 0 through 4. In this case, statement is the null statement: #include <stdio.h> for( i = 0; i < 5; cout<< i, i++) ; } for Loops and the C++ Standard The C++ standard says that a variable declared in a for loop shall go out of scope after the for loop ends. For example: for (int i = 0 ; i < 5 ; i++) { // do something } // i is now out of scope under for Scope By default a variable declared in a for loop remains in scope until the for loop’s enclosing scope ends.

It is also possible to use the scoping differences of the for loop to re-declare variables as follows: void main(){ int i = 0; // hidden by var with same name declared in for loop for ( int i = 0 ; i < 3; i++ ) {} for ( int i = 0 ; i < 3; i++ ) {} } This more closely mimics the standard behavior of a variable declared in a for loop, which requires variables declared in a for loop to go out of scope after the loop is done. When a variable is declared in a for loop, the compiler internally promotes it to a local variable in the for loop’s enclosing scope even if there is already a local variable with the same name. Like the while-loop, this loop repeats statement while condition is true. But, in addition, the for loop provides specific locations to contain an initialization and an increase expression, executed before the loop begins the first time, and after each iteration, respectively. Therefore, it is especially useful to use counter variables as condition. It works in the following way: initialization is executed. Generally, this declares a counter variable, and sets it to some initial value. This is executed a single time, at the beginning of the loop.

condition is checked. If it is true, the loop continues; otherwise, the loop ends, and statement is skipped, going directly to step 5. statement is executed. As usual, it can be either a single statement or a block enclosed in curly braces { }. increase is executed, and the loop gets back to step 2. the loop ends: execution continues by the next statement after it. Here is the countdown example using a for loop: #include <iostream.h> OUTPUT : 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, liftoff! The three fields in a for-loop are optional. They can be left empty, but in all cases the semicolon signs between them are required. For example, for (;n<10;) is a loop without initialization or increase (equivalent to a while-loop); and for (;n<10;++n) is a loop with increase, but no initialization (maybe because the variable was already initialized before the loop). A loop with no condition is equivalent to a loop with true as condition (i.e., an infinite loop).

Because each of the fields is executed in a particular time in the life cycle of a loop, it may be useful to execute more than a single expression as any of initialization, condition, or statement. Unfortunately, these are not statements, but rather, simple expressions, and thus cannot be replaced by a block. As expressions, they can, however, make use of the comma operator (,): This operator is an expression separator, and can separate multiple expressions where only one is generally expected. For example, using it, it would be possible for a for loop to handle two counter variables, initializing and increasing both: for ( n=0, i=100 ; n!=i ; ++n, –i ) { // whatever here… } This loop will execute 50 times if neither n or i are modified within the loop: n starts with a value of 0, and i with 100, the condition is n!=i (i.e., that n is not equal to i). Because n is increased by one, and i decreased by one on each iteration, the loop’s condition will become false after the 50th iteration, when both n and i are equal to 50.

Enough with the theory, now let us see some program based on for iteration statement. Program 1 : C++ Program to add 10 numbers entered by the user using FOR loop Instruction : The program is quite simple. You have already seen this program in while and do-while section ( if you have ). This program adds 10 natural numbers entered by the user using for loop. Solution : cout<<“Addition of 1st “<<n<<” even numbers are : “<<sumeven<<endl; getch(); } Program 3 : C++ Program to calculate sum of even and odd numbers uptil “n” using FOR loop Instruction : This program calculate the sum of even and odd numbers until it reaches the value of “n”. For eg : if n=10, then sumodd=1+3+5+7+9=25 and sumeven=2+4+6+8+10=30. Here sumodd and sumeven are the variables used to store the addition of odd and even numbers respectively. This program also uses if-else conditional statement. This program is same as above with slite difference.

Solution : cout<<“Addition of even natural numbers before “<<n<<” are : “<<sumeven; getch(); } Program 4 : C++ Program to calculate addition of 1st 10 odd and even numbers using FOR loop Instruction : This program calculate the sum of first 10 even and odd natural numbers. For eg : if n=10, then sumodd=1+3+5+7+9=25 and sumeven=2+4+6+8+10=30. Here sumodd and sumeven are the variables used to store the addition of odd and even numbers respectively. This program also uses if-else conditional statement. This program is same as above with 10 instead of n. Solution : cout<<“Addition of 10 numbers entered by the user using do…while loop is : “<<sum; cout<<“ndo – while loop ends…n”; getch(); } Program 6 : C++ Program to calculate the Factors of the given number also print the total numbers of Factors using FOR loop Instruction : In these particular program we are finding the factors of the number given by the user. We are also calculating the total number of factors for the given number.

For e.g.: If user enters number 10 then we are displaying its factors i.e. 1, 2, 5, 10 and also total number of factors i.e. 4. Solution : getch(); } Program 7 : C++ Program to check whether the given pair of numbers is Amicable or Not Instruction : Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number that equals the sum of its own proper divisors, in other words a number which forms an aliquot sequence of period 1. Numbers that are members of an aliquot sequence with period greater than 2 are known as sociable numbers.

For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220. The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368) In these particular program we are accepting two numbers num1 and num2 to check whether they are Amicable number or not. We have declared 2 variables sum1 and sum2 which are used to store the addition of factors of num1 and num2 respectively.

at the end we are checking the condition if((num1==sum2) && (num2==sum1)) which determines the given pair of numbers is a Amicable Pair or not. Solution : Adi Cox Number Dictionary.pdf - DocDroid docdroid Numbers are my friends. if I was a number I would want to be happy and lucky. The prime number 367 is both lucky and happy but being a prime number means that it is deficient and it is stifled in base 7. _____________________________________________________________________ Any number that is the root of a non-zero polynomial with rational coefficients is an algebraic number. Amicable The smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.

{(220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), ...} A natural number greater than 1 that is not a prime number is called a composite number. The n-th decagonal number is given by the formula d = 4n^2 - 3n. So the 5th decagonal number is: 4[5]^2-3[5] = 85 {0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, ...} Dodecagonal The dodecagonal number for n is given by the formula 5n^2 - 4n where n > 0.

So the 3rd dodecagonal is 5[3]^2-4[3] = 45-12 = 33 {1, 12, 33, 64, 105, 156, 217, 288, 369, ...} E An even number is an integer of the form n = 2k, where k is an integer. {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...} F The nth factorial is the product of all n digits: 0!=1, 1!=1, 2!=2x1=2, 3!=3x2x1=6, 4!=4x3x2x1=24, 5!=5x4x3x2x1=120, ...

{1, 1, 2, 6, 24, 120, 720, ...} Figurate A member of the subset of the sets above containing only triangular numbers, square numbers, pentagonal numbers, hexagonal numbers, heptagonal numbers, octagonal numbers, nonagonal numbers, decagonal numbers, dodecagonal numbers, pyramidal numbers, and their analogs in other dimensions. Friendly The smallest friendly number is 6, forming for example the friendly pair 6 and 28 with abundancy s(6) / 6 = (1+2+3+6) / 6 = 2, the same as s(28) / 28 = (1+2+4+7+14+28) / 28 = 2.

H A happy number goes to one when the digits of the numbers are squared and added together: 2^2+3^2=13 --> 1^2+3^2=10 --> 1^2+0^2=1 {1, 7, 10, 13, 19, 23, 28, ...} Harshad The number 18 is a Harshad number in base 10, because the sum of the digits 1 and 8 is 9 (1+8=9), and 18 is divisible by 9. 7864 is a Harshad number in base 2: 7864=1111010111000. The sum of the digits equal 8 and 7864/8=983. Heptagonal The n-th heptagonal number is given by the formula (5n^2-3n)/2.

So the 7th heptagonal number is: (5[7]^2-3[7])/2 = (5x49-21)/2 = 112 {1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, ...} Hexagonal The formula for the nth hexagonal number is: 2n^2-n. So the 5th hexagonal number is: 2[5]^2-[5] = 2x25-5 = 45 {1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, ...} I An integer is a whole number which also includes negative numbers. {...

-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10, ...} A natural number is a non negative integer: {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, ...} The nonagonal number for n is given by the formula: (n(7n-5))/2 So the 6th nonogonal number is: ([6](7[6]-5))/2 = (6x37)/2 = 111 {1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, ...} O The octagonal number for n is given by the formula 3n2 - 2n, where n > 0.

The first few octagonal numbers are: {1, 8, 21, 40, 65, 96, 133, 176, 225, 280, ...} Odd An odd number is an integer of the form n = 2k + 1 where k is an integer. {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, ...} P Is given by the formula: pentagonal number p = (3n^2-n)/2 for n greater than 1.

So the fifth pentagonal number is: (3[5]^2-[5])/2 = (75-5)/2 = 35 {1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, ...} Perfect The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. {6, 28, 496, 8128, ...} These are numbers that can be represented as dots that are arranged in the shape of a regular polygon see: Triangular, Square, Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal, Decagonal, Dodecagonal. A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Q 1264460 = 2^2.5.17.3719, the sum of the proper divisors = 1547860 1547860 = 2^2.5.193.401, the sum of the proper divisors = 1727636 1727636 = 2^2.521.829, the sum of the proper divisors = 1305184 1305184 = 2^5.40787, the sum of the proper divisors = 1264460 Square Is an integer that is a product of itself. Examples are: 1x1=1, 2x2=4, 3x3=9, 4x4=16, 5x5=25, 6x6=36, 7x7=49, 8x8=64, ...

{1,2,4,9,16,25,36,49,64,81,100,121,144, ...} A number is stifled when the digits of that number equals the base of that number: The number 10 is stifled in base 4: 2x4^1 + 2x4^0 = 10 so 22 in base 4 is equal to ten in base 10 as 2+2=4. All strange numbers are prime. Every single digit prime number is strange. A number with two or more digits is strange if, and only if, the two numbers obtained from it, by removing either its first or its last digit, are also strange. {2, 3, 5, 7, 23, 37, 53, ...} Surd When we can not simplify a number to remove a square root (or cube root, or nth root) then it is a surd. ie x^1/n. T A figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron.

The nth tetrahedral number is the sum of the first n triangular numbers. {1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...} Transcendental Any real or complex number that is not algebraic is transcendental. Examples include e and pi. Triangular A triangular number is the sum of n numbers starting from zero. So some examples are: 0, 0+1=1, 0+1+2=3, 0+1+2+3=6, 0+1+2+3+4=10, ...

{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, ...} U An unhappy number or a sad number is a number that is ot happy. see happy numbers. Special Numbers Perfect Numbers The Pythagoreans produced a theory of numbers comprised of numerology and scientific speculation. In their numerology, even numbers were feminine and odd numbers masculine. The numbers also represented abstract concepts such as 1 stood for reason, 2 stood for opinion, 3 stood for harmony, 4 stood for justice, and so on. Their arithmetica had a theory of special classes of numbers. There were �perfect� numbers of two kinds.

The first kind included only 10, which was basic to the decimal system and the sum of the first four numbers 1 + 2 + 3 + 4 = 10. The second kind of �perfect� numbers were those equal to the sum of their proper divisors. A perfect number is a positive integer that is equal to the sum of it divisors. However, for the case of a perfect number, the number itself is not included in the sum. The Greeks called a number such as 6 or 28 a �perfect� number because the sum of the proper divisors in each case is equal to the number; the proper divisors of 6 are 1, 2, and 3, and their sum is 6. Although perfect numbers are regarded as arithmetical curiosities, their study has helped to develop the theory of numbers. Euclid proved that a number n of the form (2n-1)*2n-1 is a perfect number if the factor 2n-1 is prime.

For example, if n assumes the value 2, 3, 5, or 7, the expression 2n-1 takes on the value 3, 7, 31, or 127, all of which are prime. For these values of n we obtain the perfect numbers 6, 28, 496, and 8,128. The Neoplatonists Nicomachus of Gerasa and Iamblichus of Chalcis listed these perfect numbers and concluded that they follow a pattern: They alternately end in a 6 or an 8, and there is one perfect number for each interval from 1 to 10, 10 to 100, 100 to 1,000 and 1,000 to 10,000. They conjectured that both parts of the pattern would continue, but in this they were wrong. The fifth perfect number, which was discovered in the fifth century, corresponds to n = 13 and is 33,550,336, with eight digits rather than six.

In addition, the sixth perfect number, like the fifth, ends with a six. In 1961, the twentieth perfect number was found. It contains 2,663 digits in the decimal representation and corresponds to the case where n = 4,423. Today, thirty-seven perfect numbers are known. The prime for the largest of these is 23,021,377, which is 909,526 digits in length, and the largest perfect number is 1,819,050 digits in length It is not known whether there are an infinite number of perfect numbers. In 1757, Leonhard Euler probed that every even perfect number must be of Euclid�s form. It has also been proven that every even perfect number must end in six or eight and if it ends in six, the digit preceding it must be odd. No one has as yet discovered an odd perfect number, but it is known that none exist below 1020. Contributed by Kristen Shelton Calinger, Ronald, Classics of Mathematics, Prentice Hall, New Jersey 1995.

Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., New York 1989. The Nature of What is (Pi)? (Pi) is the ratio of the distance around a circle to the circles' diameter. Mostly (Pi) is written as a fraction: (distance around a circle)/(distance across and through the center of a circle) or more simply C/D The result of this fraction is a famous nonending, nonrepeating infinite string of decimals that mathematicians have been studying ever since Archimedes (287-212 B.C.E.). Archimedes is credited with providing a method of calculating (Pi). This number goes something like this: 3.14159265358979323 and so on. Your calculator ( or most calculators) go ten places. This is considered more than enough to ensure accuracy in measurements. Measuring is one of the main things you can do with this number. What does this number mean? What mathematicians discovered is the existence in nature of ratios. A ratio is a relationsip between two things.

For example 1/2 is a ratio that gives us information about how the 1 and 2 are related. In this case 1 is divided by 2. Not only do these relationhisp exist but some of them 'always' have the same answer. They don't change. In other words the ratio of a circles' diameter to its circumference is always the same. It doesn't matter what these two numbers are. Their ratio is always equal to that very long number with all the decimals. What is (Pi) good for? You can find the area of a circle, the circumference of a cirlce, and the volume of a cylinder using the value of (Pi). Remember finding the area of a quadrilateral is the length(l) times the width(w). l x w In the figure below the area is equal to the sum of the 'squares' found in the figure. Now try to find the number of 'squares' in a circle. How do you account for shaded areas that are not square? Some math people call these shapes lunes. You can calculate area much easier by using your formula for (Pi): (1/2)*(Pi)*r2. In this case the radius equals 1.

The value for (Pi) is 3.14. The formula now looks like this: (1/2)(3.14)(12) = area of the circle. History of (Pi) Since (Pi) is a nonending,nonrepeating, infinite decimal it has stood as a monument to futility and utter uselessness. Various people have sat around calculating (without calculators, calculus, or even algebra) this number.: Ptolemy Prime Numbers A natural number that possesses only two factors, itself and 1, is called a prime number. You may think that it would be fairly easy to figure out all of these numbers. Well, I am sure that you could find the prime numbers within the first 20 natural numbers without much trouble, and you may be able to go even further without the use of a calculator. However, it does become much harder to figure out all of the prime numbers, especially in your head. It was an early Greek mathematician that is most famous for his work with prime numbers. His name was Eratosthenes. It was this Greek mathematician that created what is known as the Sieve of Eratosthenes.

This Sieve allows anyone to find all of the prime numbers quite easily. This is how the Sieve works . . . Eratosthenes wrote all of the natural numbers down and proceeded to sieve out all of the numbers that were not primes. The number one has only one factor, so it is not a prime number. We move to number two which is a prime number. Strike every second number after the number two (or multiple of two). Move to the next prime number. It is three. Now we strike every third number after the number three (or multiple of three) and so on. We continue to strike out every nth number after the number n. Now you are probably wondering just how many prime numbers there are. Well, there are infinitely many primes. In other words, there are more than we could ever count. We know that there is an infinite number of primes because if you were to multiply all of the known primes together and add 1, then you would get a number that must be divisible by at least one new prime number. You can keep doing this forever and never find the end of the prime number list. One more interesting thing about prime numbers.

Prime numbers are considered the "building blocks" of the natural numbers because every single natural number, excluding the number 1, is either a prime number or a product of prime numbers. In other words, every number, for example, 160 is either a prime or can be factored into a product of primes. To factor 160, we observe that 160 is even and hence is divisible by 2 (which happens to be the only even prime), so we divide 160 by 2, giving 160 = 2*80. Now 80 is even and nece divisible by 2, so 80 = 2*40 and 160 = 2*2*40. But 40 is also divisible by 2, 40 = 2*20, and 160 = 2*2*2*20. Continuing to divide by 2 until we can no longer do so, we get 160 = 2*2*2*2*2*5.

Now 5 is a prime, so we would write 160 as a product of primes like this: 2*2*2*2*2*5 = 25*5. If we were to multiply all of these prime numbers, the answer would be 160. Thus, since all numbers other than 1 are prime numbers or products of prime numbers, we can use a process like the above to find the prime divisors of any number. Enjoy your prime number hunting! Contributed by Courtney Ast Amicable Numbers Throughout history there have been many different interesting numbers or types of numbers. One of these types is amicable numbers. Amicable numbers are a pair of numbers with the following property: the sum of all of the proper divisors of the first number (not including itself) exactly equals the second number while the sum of all of the proper divisors of the second number (not including itself) likewise equals the first number.

For example let's show that 220 & 284 are amicable numbers: First we find the proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 If you add up all of these numbers you will see that they sum to 284. Now find the proper divisors of 284: 1, 2, 4, 71, 142 These sum to 220, and therefore 220 & 284 are amicable numbers. The set of 220 and 284 was the first known set of amicable numbers. Pythagoras discovered the relationship and coined the term amicable because he considered the numbers to be a symbol of friendship. No other pairs were known until 1636 when Fermat discovered 17,296 and 18,416 as a second pair.

This pair was actually discovered over three hundred years earlier by the Arab mathematician al-Banna, but it was never known in the West until Fermat's findings. Then in 1638, Descartes discovered a third pair of 9,363,584 and 9,437,056. The First Thirteen Amicable Pairs 1 79,750 88,730 It wasn't until 1747 when Leonhard Euler turned his attention to amicable numbers that progress began to take place. Euler was able to produce 58 pairs over the next three years. How did he find so many, so quickly? Euler developed a formula that would produce amicable pairs. The only problem was that the formula didn't generate every amicable pair. Today there are over 5000 known pairs, the largest of which was found by Mariano Garcia on October 4, 1997 contained 4829 digits in each pair. There are still many questions left to be answered about amicable numbers. Are there an infinite amount of pairs? Do the pairs always appear so that they are both even or both odd?

What about the existence of amicable triples or quadruples? These are all questions that mathematicians are searching & researching today. Who knows, maybe you could discover some of these answers and put yourself in the history books alongside Euler and Fermat. Contributed by Damian Smithhisler The Golden Ratio: Phi The ratio of 1:1.61803 has unique and storied history. It is also know as Phi, named after the Greek sculptor Phidias. The ratio is found in nature, art, architecture, poetry, music, and of course math. As stated, the Golden Ratio is also known as Phi. Phidias lived from c. 500 - 432 BC. He incorporated the ratio into many of his sculptures, one of which is the ancient Greek temple, the Parthenon. This was a rectangle shaped temple whose sides are in the Agolden proportion@. The number Phi is as unique as the number pi. It is a number whose decimal never ends and never has a pattern repeat itself. Also, there are only two numbers that remain the same when you square them, 0 and 1. To fine the value of Phi squared you simple add one to Phi.

There are only two numbers that have this property. It is Phi and a number closely related to it: 0.61803. This number is also known as phi (note the lower case p). Phi is supposedly the >most pleasing= rectangle proportion. It is found in not only Greek architecture, but other architecture as well. A couple examples are the pyramids and the United Nations building. The Golden Ratio is also seen in nature. It can be found in many different ways on the human body. For instance, measure from the top of the head to the base of the neck and also from the base of the neck to the belly button. You will find this ratio is close to the Golden Ratio. Other examples can be found in flowers, ferns, eggs, and sea shell to name a few. It is found in art, also. Many books about painting will point out that it is better to position objects to Aabout one-third@ of the way across and not in the center of the picture. This seems to make the picture more pleasing to the eye. Another interesting thing about the Golden Ratio has to do with the Fibonacci sequence.

If you find the ratio of two successive numbers in the sequence you will find as the numbers get larger the ratio gets closer and closer the Phi. For instance, 5/3 = 1.667. 233/144 = 1.6181. 610/377 =1.6181. Examples of the Golden Ratio can be seen everywhere. It is a unique number that has some very interesting properties. I have only touched on the surface of what is out there about it. Please check some of the web sites listed below to find out more about the Golden Ratio or Phi. Contributed by James Means Animals and Numbers Can animals count or even have a concept of numbers? Many specialists in animal behavior have conducted experiments that have shown animals to possess an apparent sense of rudimentary perception of quantities. This is sometimes called the "number " sense. The number sense allows an animal to determine the differences in size between two small collections of similar objects. It also determines that a collection isn't the same after some objects are removed. Mothers of domestic animals and other animals have shown that they can definitely determine and perceive when one of their young is missing from the group.

Birds have shown that they can be trained to determine the number of seeds in different piles of seeds up to the number five. An example of their ability to count up to five is contained in the story of the crow. In this story, a squire was very frustrated with a particular crow. This crow had made its' nest in the watchtower of the squire's estate. The squire continuously tried to surprise the crow, but had no such luck. Each time the squire approached the nest, the crow would leave. It would not return until the squire had left. From a distance the bird would watch for the man's departure. With this in mind, the squire came up with a plan to fool the bird. He took another man into the tower with him. Hoping to outwit the bird, one man left the tower. However, the bird would still not return to the tower until both men had departed. This experiment was repeated in the succeeding days with two, three, and even four men. Yet, the crow was not to be fooled. Finally, five men were sent into the tower. As before, all entered the tower. Only one remained as the other four left the area.

At this point, the crow lost count and returned to it's nest. Not being able to distinguish above the number four cost the crow it's life, as the squire rid the tower of the crow forever. Birds are not the only creatures with a concept of numbers. The behavior of insects, such as the solitary wasp, indicates some basic mathematical understanding. The mother wasp lays her eggs in individual cells and provides each egg with a number of live caterpillars, on which the young will feed on when they hatch. The number of caterpillars is different among species, but it is always the same for each sex of eggs. The male is smaller, so the mother supplies him with only five caterpillars. The female, who is large, receives ten caterpillars in her cell. This shows that the mother can not only distinguish which cell contains a male or a female, but she can also distinguish between the numbers five and ten in the caterpillars she is providing. Other animals, such as animals in circuses, have been hypothesized to have mathematical talents. Animals such as horses and elephants are displayed to have the abilities to add, subtract, and count.

However, these talents are more of a trained response triggered by a cue from the trainer. The skills demonstrated are not an innate response, as seen with the crow and the wasp. Contributed by Lloyd Holt Ideal & Irrational Numbers The Pythagoreans were people who followed the mathematician, Pythagoras, best known for the right triangle theorem which is named after him. They believed that the universe could be explained by numbers. They were especially aware of the numbers 1,2,3,4 which they called the tetractys. They swore to the tetractys and saw "fourness" in many things. The geometric elements point, line, surface and solid are examples of fourness. To Pythagoras, numbers had personalities - masculine or feminine, perfect or incomplete, beautiful or ugly. Ten was the best: containing one, two, three, and four - and written in dot formation it formed a perfect triangle. Because the sum of 1,2,3 and 4 is 10; ten was considered the ideal number. Whereas four represented nature, they believed that ten represented the universe. They used the eight planets which had been discovered at this time plus two invented planets to support this belief.

The Pythagoreans were unable to see these two planets because the part of the Earth where they lived "faced away from these planets." Pythagoras looked at ratios of whole numbers to explain nature. Upon his death, his followers used the method of indirect proof to show that the square root of 2 cannot be written as a ratio of two whole numbers. In using this proof, irrational numbers - numbers which cannot be written as the ratio of two whole numbers - were discovered. PROOF: Fibonacci Numbers The Fibonacci Number Sequence was first presented in Leonardo Pisano's book, "Liber abaci" or "Book of Calculating". It is a sequence that I find to be very fascinating, and suprisingly it is a part of every day nature. The Fibonacci sequence can be found in sea shell spirals, branching plants, petals on flowers, and in pine cones. I will explain this sequence to you in 3 different ways: the basic sequence, the rabbit problem, and the bees.

The Basic Sequence The first twenty numbers of the sequence are as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181 The numbers are obtained by adding two numbers to get the next. For example: 0 + 1 = 1 The History of Zero In today's modern mathematics, we have become accustomed to zero as a number. It's hard to believe that most ancient number systems didn't include zero. The Mayan civilization may have been among the first to have a symbol for zero. The Mayas flourished in the Yucatan peninsula of Mexico about 1300 years ago. They used the as a placeholder, in a vertical place-value system. It is considered one of their cultures greatest achievements. The ancient Egyptians, Romans, and Greeks alike had no symbol for zero. In Greek geometry, zero and irrational numbers were impossible.

The Greeks made great strides in mathematics, but it was all done with a number system without zero. The Greek astronomer Ptolemy (ca. A.D. 150) was the first to write a zero at the end of a number. For this he used a circular symbol. In ancient Babylonian history there was no use of the zero. In the later Babylonian or during the Seleucid period a special symbol, which was also used as a separation mark between sentences, came into use for a zero. There's a definite possibility that the Babylonians used this mark for a zero within a number, as early as the end of the eighth century B.C. Up until the time of Aristotle, there seems to be no evidence that the Babylonians ever regarded zero as a number. Aristotle discussed division by zero in connection with speed through a vacuum. Throughout the Dark Ages, Western mathematics was held back by the Roman's traditional numbering system. The first to think differently was Leonardo Fibonacci. He was a merchant's son, born in the Italian city-state Pisa, late in the twelfth century. In Pisa, he studied the work of Euclid and other Greek mathematicians.

When he was still a boy, he moved to the Muslim city of Bugia, in North Africa. There he examined leather and furs before they were shipped back to Pisa. Leonardo got an education in Arabic culture as he traveled around the Mediterranean to Constantinople, Egypt and Syria. He recognized that the Hindu-Arabic numerals, the numerals we use today, were superior to the Roman numerals he had grown up with in the West. In the sixth century, mathematicians in India developed a place-value system. They introduced the concept of zero to keep their symbols in their proper places. In the seventh century, Hindu scholars introduced to Islam the ideas of zero and place-value. These ideas spread rapidly throughout the Arabic world. Six centuries later, Fibonacci was so impressed with the ease of Hindu-Arabic numerals that he wrote a book entitled Liber abaci. The Pisan local merchants, the trading class, ignored Fibonacci's book. They were wallowing in prosperity and did not want to be bothered with giving up Roman numerals and adopting a zero. Ferbonacci's mathematician friends liked the new number system and slowly over time gave up the Roman numerals.

By the fifteenth century, the numerals were showing up on coins and gravestones. Western mathematics had emerged from the Dark Ages, and was flourishing into a new number system with a zero, the Hindu-Arabic numerals. The immediate advances in mathematics after that time are proof of the importance of, the zero. Contributed by Pam Nye

Question: The following are the first few examples of what: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564) & (6232, 6368)?
Answer: Amicable numbers