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Types of Numbers, Part IV

Compiled by William Tappe

PERFECT NUMBERS

 

The definitions of a perfect number vary. A few of the variations follow:

   1-- A perfect number, N, is a number equal to the sum of all its proper divisors (including 1).

   2-- A number, N, is perfect if it is the sum of all its factors/divisors, not including the number itself.

   3-- If the sum of the divisors of N is equal to N, N is defined as a perfect number.

   4-- A number is said to be perfect if it is equal to the sum of its proper divisors.

   5-- A perfect number is a number equal to the sum of its divisors (not including itself).

   6-- A perfect number is a number that is equal to the sum of its aliquot parts.

       Confusing? Perhaps. In the language of the Greek mathematicians, the divisors of a number N were defined as any whole numbers smaller than N that, when divided into N, produced whole numbers. The divisors thus derived were called aliquot divisors, or aliquot parts, and did not include the number itself. Thus, a perfect number was defined as a number that is equal to the sum of its aliquot parts (divisors). (Proper divisors later became synonymous with aliquot parts.) In order to obtain the sum sa(N) of the aliquot parts of a number, one must diminish the sum, s(N), of all its divisors (including the number itself) by the so called improper divisor N, such that sa(N) = S(N) - N. The condition for a perfect number may then be defined by sa(N) = N, or equivalently, s(N) = 2N. The sum of the aliquot parts equaling the number is the more traditionally accepted definition. 

 

The number 6 is a perfect number, since its positive aliquot parts (proper divisors) are 1, 2, and 3 and 1+2+3 = 6. At present, there are over 30 known perfect numbers, all even. It is not known if there are any odd perfect numbers; none has been found, but it has not been proved that one cannot exist. All even perfect numbers derive from 2^(p-1)(2^p - 1), where p is any positive integer, exceeding unity, that results in (2^p -1) being a prime number. The primes of the form (2^p - 1), where p is a prime, are called Mersenne primes after the French mathematician who announced a list of perfect numbers in 1644. The known values of p that yield Mersenne primes and corresponding perfect numbers are: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1,279, 2,203, 2,281, 3,217, 4,253, 4,423, 9,689, 9,941, 11,213, and 19,937. Though there are undoubtedly many more beyond p = 19,937, their size grows rapidly as p increases. The 15th Mersenne prime, (2^1,279 - 1) has 386 digits. The first eight perfect numbers are  6, 28, 496, 8128, 33,550,336, and 8,589,869,056, 137,438,691,328, and 2,305,843,008,139,952,128, having p's of 2, 3, 5, 7, 13, 17, 19, and 31. All perfect numbers end in either a 6 or an 8.

 

A perfect number, or any number, of the form N = p^a(q^b)r^c...., has a total number of divisors equal to D = (a + 1)(b + 1)(c + 1). The number of aliquot divisors is therefore Da = D - 1.

 

A perfect number, N, may be broken down into its prime factors p^a(q^b)r^c..., from which the sum of its divisors is s(N) = 2N = [1+p+p^2+...p^a]x[1+q+q^2+...q^b]x[1+r+r^2+...r^c] = [{p^(a+1) - 1}/(p -1)]x[{q^(b+1) - 1}/(q - 1)]x[{r^(c+1) - 1}/(r - 1)]. The sum of its aliquot divisors is therefore sa(N) = s(N) - N.

 

Another peculiar property of perfect numbers is the fact that the sum of the reciprocals of the divisors of the number add up to 2 and every perfect number is the sum of consecutive odd cubes (except 6). For example, 6 = 1 + 2 + 3 and 1/1 + 1/2 + 1/3 + 1/6 = 2. 28 = 1 + 2 + 4 + 7 + 14 and 1/1 + 1/2 + 1/4 + 1/7 + 1/14 + 1/28 = 2 and 1^3 + 3^3 = 28. Similarly, 496 = 1^3 + 3^3 + 5^3 + 7^3.

 

The net result of successively adding the digits of a perfect number is always 1, with the exception of  6.

Example: From 8128, 8+1+2+8 = 19 and 1+9 = 10 and 1+0 = 1.

From 33,550,336, 3+3+5+5+0+3+3+6 = 28 and 2+8 = 10 and 1+0 = 1.

 

Every perfect number (except 6) of the form 2^a[2^(a + 1) - 1] is the sum of the cubes of the first 2^(a/2) odd numbers.

Perfect number......................28......496.........33,550,336

a..............................................2.........4.................12

No. of odd #'s cubed...............2.........4.................64

Odd numbers cubed............1,3....1,3,5,7....1,3,5,...>127

 

A semi-perfect number is a number that is equal to the sum of some of its aliquot parts or proper divisors. The smallest semi-perfect number is 12 which is the sum of 2, 4, and 6. 18 is the next with 18 = 3 + 6 + 9. 24 is another with 24 = 4 + 8 + 12 or 1 + 2 + 3 + 4 + 6 + 8.

 

(See semi-perfect, multiply-perfect, quasi-perfect, deficient, least deficient, abundant, super abundant)

 

PERFECT NUMBERS (Almost)

 

Powers of 2 are often referred to as almost perfect numbers. This is due to the fact that the aliquot parts of any 2^n power sum to 2^n - 1. Of course, in reality, the powers of 2 are deficient numbers but only by 1. It remains to be seen whether there are any odd numbers whose aliquot parts sum to one less than the number itself.

 

PERIODIC NUMBERS

 

Periodic numbers are numbers where the same set of digits repeat themselves. Numbers such as 729472947294, 27272727, 318318318, .14141414..., .259259259..., .257825782,578..., are examples. The decimal form of periodic numbers are more often called repeating decimals.

....................................................................................................................__...___..........____

Simpler representations of the decimal repeating decimal numbers are .14, .259, and .2578, the line above the repeating digits meaning that they go on forever.

 

PERMUTABLE PRIME

 

A permutable prime is a prime number of two or more digits, that remains prime with every possible rearrangement of the digits, For example, 79 and 97 are both prime. Consider the number 337, which can be rearranged to 373 and 733, all three of which are primes. The digits 2, 4, 6, 8 or 5 cannot be in a permutable prime.

              

PERSISTENT NUMBERS

 

Persistent numbers are numbers, the sequential product of whose digits, eventually produces a single digit number. For example, take the number 764. 7x6x4 = 168; 1x6x8 = 48; 4x8 = 32; and finally, 3x2 = 6. The number of multiplication steps required to reach the single digit number from the given number is referred to as the "persistence" of the starting number. The challenge often posed is in finding the smallest possible numbers with given levels of persistence. For instance, what might be the smallest number with a persistence of 1? My first thought would be 11 where 1x1 = 1. How about the smallest number with persistence of 2? My choice would be 26 where 2x6 = 12 and 1x2 = 2. As for a persistence of 3, I would opt for 39 where 3x9 = 27, 2x7 = 14, and 1x4 = 4. See how many smallest numbers you can find with higher levels of persistence.

 

POLYGONAL NUMBERS

 

Polygonal numbers, often referred to as figurate numbers, derive their names from the numerous geometric arrangements of dots, circles, spheres, etc., that they form.

 

The number of dots, circles, etc., that can be arranged in a line is called a linear number. These numbers, L1, L2, L2, etc., are simply the counting numbers, 1, 2, 3, 4, 5.....n.

 

The number of dots, circles, spheres, etc., that can be arranged in an equilateral or right triangular pattern is called a triangular number. The 10 bowling pins form a triangular number as do the 15 balls racked up on a pool table. Upon further inspection, it becomes immediately clear that the triangular numbers, T1, T2, T3, T4, etc., are simply the sum of the consecutive integers 1-2-3-4-.....n or Tn = n(n + 1)/2, namely, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66,78, 91, etc.

    The sum of a series of triangular numbers from 1 through Tn is given by S = (n^3 + 3n^2 + 2n)/6.

 

The number of dots, circles, spheres, etc., that can be placed in a square array is called a square number.    Obviously, these square polygonal numbers follow the series of squares S1, S2, S3, S4, 3tc., 1, 4, 9, 16, 25, 36, 49, 64, 81, etc.

   The nth square number is Sn = n^2.

   The sum of series of square numbers from 1 through Sn is given by S(1>n) = (2n^3 + 3n^2 + n)/6.

 

After staring at several triangular and square polygonal number arrangements, one can quickly see that the 1st and 2nd triangular numbers actually form the 2nd square number 4. Similarly, the 2nd and 3rd triangular numbers form the 3rd square number 9, and so on. By inspection, one can see that the nth square number, Sn, is equal to Tn + T(n - 1) = n^2. This can best be visualized from the following:

.........Tn -        1...3...6...10...15...21...28...36...45...55...66...78...91

.........T(n - 1)........1...3....6....10...15...21...28...36...45...55...66...78

.........Sn.........1...4...9....16...25...36...49...64...81..100.121.144.169

 

A number cannot be triangular if its digital root is 2, 4, 5, 7 or 8.

 

The numbers 1 and 36 are both square and triangular. Some other triangular squares are 1225, 41,616, 1,413,721, 48,024,900 and 1,631,432,881. Triangular squares can be derived from the series 0, 1, 6, 35, 204, 1189............Un where Un = 6U(n - 1) - U(n - 2) where each term is six times the previous term, diminished by the one before that. The squares of these numbers are simultaneously square and triangular.

   

Pentagonal numbers derive from triangular numbers and square numbers by Pn = Tn + S(n + 1). See below.

 

Triangular Numbers

Rank....1........2...............3.....................4.............................5....................6.....7....8.....9

...........O.......O...............O....................O............................O

..................O...O.........O...O..............O...O......................O...O

.................................O...O...O........O....O....O.............O....O....O

.....................................................O....O...O....O......O.....O...O....O

...............................................................................O....O....O....O....O

 

Total....1........3.................6....................10..........................15..................21...28...36...45...etc.

 

Square Numbers

    Square numbers are the sum of the similar spots, etc., in a square.

 

Order...1........2...............3....................4.........................5.......................6.....7.....8....9

..........O....O...O........O...O...O.....O...O...O...O.......O...O...O...O...O

.................O...O........O...O...O.....O...O...O...O.......O...O...O...O...O

.................................O...O...O......O...O...O...O.......O...O...O...O...O

......................................................O...O...O...O.......O...O...O...O...O

..................................................................................O...O...O...O...O

 

Total....1........4...............9...................16..........................25...................36...49...64...81...etc.

 

Pentagonal Numbers

Pentagonal numbers derive from triangular numbers and square numbers by Pn = Tn + S(n + 1). The nth pentagonal number has n sides on the major pentagon it represents. See below.

 

Order...1..  ......2............................3...............................4

...........O..  .....O............................O..............................O

.............  ...O.......O..................O.......O....................O.......O

...............  ...O...O...............O....O...O....O...........O...O...O...O

...............  .............................O.............O.........O...O...........O...O

................  ..............................O...O...O............O....O..O..O....O

................  ..........................................................O..................O

...............  .............................................................O...O...O...O

 

..........1...........5............................12.............................22  

Sides 1...........2.............................3...............................4

 

   I'll let you work out the 5th rank pentagonal number from the next picture of from the general expressions below.

 

The polygonal numbers derive from the addition of the first n terms of unique arithmetic progressions, all starting with 1.

Counting numbers, Cn, derive from 1 + 1 + 1 + 1 + 1 + 1 + ............... = 1, 2,  3,   4,   5,   6,    7......

Triangular numbers, Tn, derive from 1 + 2 + 3 + 4 + 5 + 6 + ...............= 1, 3,  6,  10, 15,  21,  28 .....= n(n+1)/2

Square numbers, Sn, derive from 1 + 3 + 5 + 7 + 9 + 11+...................= 1, 4,  9,  16, 25, 36,  49 ......= n^2

Pentagonal numbers, Pn, derive from 1 + 4 + 7 + 10 + 13 + 16 + ...... = 1, 5, 12,  22, 35, 51,  70 .....= n(3n-1)/2

Hexagonal numbers, Hxn, derive from 1 + 5 + 9 + 13 + 17 + 21 + .......= 1, 6, 15, 28, 45, 66,  91.....= n(2n-1)

Heptagonal numbers, Hpn, derive from 1 + 6 + 11 + 16 + 21 + 26 + ... = 1, 7, 18, 34, 55, 81, 112.....= n(5n-3)/2

Octagonal numbers, On, derive from 1 + 7 + 13 + 19 + 25 + 31 + ...... = 1, 8, 21, 40, 65, 96, 133.....= n(3n-2)

n-gonal = n[(n-1)n - 2(n-2)]/2  where n = the order number.

 

(Note the differences between each polygonal number in the same position. That is, the differences between the 2nd numbers are all 1, the differences between the 3rd numbers are all 3, the differences between the 4th numbers are all 6, the differences between the 5th numbers are all 10, the 6th numbers 15, the 7th numbers 21, and so on, the sequence of the triangular numbers.)

 

Every octagonal number is the difference of two squares.

Example: 21 = 5^2 - 2^2; 40 = 7^2 - 3^2; 65 = 9^2 - 4^2.

 

You might now notice that the 1st and 2nd triangular numbers add up to the 2nd square number. Similarly, the 2nd and 3rd triangular numbers add up to the 3rd square number. This relationship can be expressed by Sr = Tr + T(r-1) where Sr is the square number of rank r and Tr is the triangular number of rank r. Since Tr = r(r-1)/2 and T(r-1) = (r-1)(r-1+1)/2 = (r-1)r/2, we can write Tr + T(r-1) = r(r+1+r-1)/2 = r^2 and r^2 is obviously the same as Sr.

 

In the same sense that a square number can be derived from the sum of two triangular numbers, a pentagonal number, Pn, can be derived from the sum of a square number of a particular rank and the triangular number of the previous rank or Pn = Sn + T(n-1). For instance, the 4th pentagonal number P4 = S4 + T3 = 16 + 6 = 22.   Similarly, Hxn = Sn + T(n-2).

 

Similarly, Hxn = Pn + T(n-1), Hpn = Hxn + T(n-1), On = Hpn + T(n-1), and so on.

 

Another strange relationship between triangular numbers and their squares.

...Triangular....1.......3.......6.......10.......15.......21........28

...Square.......1..... ..9......36.....100......225.....441......784

...Difference........8......27......64......125.....216......343

...Equals.....,....2^3....3^3.....4^3......5^3......6^3......7^3

 

   Thus, from the nth Triangular number, Tn, (Tn)^2 - (T(n-1))^2 = n^3.

 

The sum of the first three triangular numbers equals the fourth, T1 + T2 + T3 = T4. Would it surprise you to learn that

...T5 + T6 + T7 + T8 = T9 + T10

...T11 + T12 + T13 + T14 + T15 = T16 + T17 + T18

...T19 + T20 + T21 + T22 + T23 + T 24 = T25 + T 26 + T27 + T28 ad infinitum.

   The start of each series follows the pattern 1, 5, 11, 19, 29, 41, with the end of each series following the pattern of 4, 10, 18, 28, 40, and so on. 

 

A Triangular Triple results when three natural numbers a, b, c, with a<=b<=c, produce Ta + Tb = Tc. For example, T(14) + T(18) = T(23) where T(14) = 105, T(28) = 171, and T(23) = 276 resulting in 105 + 171 = 276.

 

Would you believe that the sum of n consecutive cubes, beginning with 1, is always equal to the square of the nth triangular number?  

 

Much more on the subject can be found in the following excellent books:

1--Recreations in the Theory of Numbers by Albert H. Beiler, Dover Publications, Inc., 1964.

2--Mathematical Recreations and Essays by W.W. Rouse and H.S.M. Coxeter, Dover Publications, Inc., 1987.

3--The Book of NUMBERS by J.H. Conway and R.K. Guy, Copernicus/Springer-Verlag, 1996.

 

POWERFUL NUMBER

 

A powerful number is any  positive integral number, N, which, if evenly divisible by a prime number, p, is also evenly divisible by p^2.Take the number 72 for instance; 72 is divisible by 2 and 4, 3 and 9, and 6 and 36. Every powerful number is of the form a^2b^3. 

 

PRIME NUMBERS

 

A prime number is

1--a positive number "p" that has but two positive divisors/factors, 1 and "p" or

2--any natural number larger than 1 that cannot be written as the product of two smaller numbers.

   (The strict interpretation of this definition aids in supporting the statement that the number one is not a prime number as it is the only number with one divisor/factor whereas a prime always has two factors.)

   Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc., are all primes, being evenly divisible by only 1 and the number itself.

 

*  The Fundamental Theorem of Arithmetic states that every positive integer/number greater than 1 is either prime or composite and can be written uniquely as a product of primes, with the prime factors in the product written in order of non-decreasing size.

 

*  Any number that has 3 or more factors/divisors is called a composite number..

 

*  The number 1 is not considered a prime number, being more traditionally referred to as the unit.

  

*  A composite number is expressible as a unique product of prime numbers and their exponents, in only one way.

   Examples: 210 = 2x3x5x7; 495 = 3^2x5x11.

 

*  A prime factor of a number is a divisor/factor of the number which also happens to be a prime number. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36 but only 2 and 3 are prime factors.

 

*  The number 1 is the only number that is a factor of all other numbers.

 

*  Any number that can be expressed as the product of two or more primes and their powers, i.e., ab, abc, ab^2c, ab^2c^3d^2, etc., where a, b, c and d are prime numbers, is composite.

 

*  Any number greater than 1 that is not a prime number must be a composite number, and is the result of multiplying primes together.

Examples: 4, 6, 12, 24, 72, etc., are composite, each being divisible by lower prime numbers.

 

*  Every number n > 1 is divisible by some prime.

 

*  With the exception of the number 2, all prime numbers are odd numbers.

   The number 2 is the only even prime, thereby making it the oddest prime.

 

*  All prime numbers of two digits or more end in 1, 3, 7, or 9. 

   Caution: There are numbers that end in 1, 3, 7, or 9 that are not prime but, in order to be a prime, it must end in 1, 3, 7, or 9.

 

*  Two integers are said to be relatively prime, or prime to each other, if their greatest common divisor is 1.

   Examples: 7 and 12 are relatively prime as their g.c.d. is 1 or (7,12) = 1.

 

*  A semi-prime number is one that has exactly two prime factors in addition to the factors of one and the number itself. For example, 6, 10, 14, 15, 21, 39, 142, 143, etc. are semi-prime numbers.  By definition, such numbers are also classified as composite numbers.

 

*  Primes differing by 2 are called twin primes.

   Examples: 3-5, 5-7, 11-13, 17-19, 29-31, 41-43, 59-61, 71-73, 101-103, 1,000,000,009,649-1,000,000,009,651.

 

*  The sums of all pairs of twin primes (except 3-5) are divisible by 12.

 

*  There is only one set of triple primes - 3, 5, and 7.

 

*  The prime number 5 is the only prime number that is both the sum and difference of two other primes.

 

*  The primes 2 and 3 are often referred to as the Siamese Twins as they are the only two adjacent prime numbers.

 

*  Every odd number is congruent to either 1 or 3 modulo(4) meaning that every odd number minus 1 or 3 is divisible by 4.

 

*  Every prime number is congruent to one of 1, 2, 3 or 4 modulo(5).

 

*  There are infinitely many primes of the form N^2 + 1. If of the form N^2 + 1, it does not mean it is automatically prime. Primes of the     form N^2 + 1 are either prime or the product of two primes.

 

*  Every prime of the form 4n + 1 can be written as the sum of two squares, 5, 13, 17, 29, 37, 41, 53, etc.

 

*  No primes of the form 4n - 1 can be written as the sum of two squares, 3, 7, 11, 43, 47, 59, 67, etc.

 

*  The number 17 is the only prime that is equal to the sum of the digits of its cube.

 

*  A prime p is the sum of two squares if, and only if, p + 1 is not divisible by 4.

 

Finding Primes

 

 

The Sieve of Eratosthenes

 

Lets find the primes between 1 and 100.

Write down the sequence of numbers from 1 to 100.

Cross out the 1.

Beginning with the 2, strike out every second number beyond the 2, i.e., 4, 6, 8, 10, etc.

Starting from the first remaining number, 3, cross out every third number beyond the 3, i.e., 3, 6, 9, 12, etc.

Starting from the first remaining number, 5, cross out every fifth number beyond the 5, i.e., 5, 10, 15, 20, etc.

Continue with the 7, crossing out every seventh number beyond the 7, i.e., 7, 14, 21, 28, etc.

Continue the process until you have reached N or 100.

The numbers remaining are the primes between 1 and 100, namely 2, 3, 5, 7, 11, 13,17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. By definition, all the others are composite numbers.

 

*  The largest prime discovered as of November 2003 is 2 to the 20,996,011 power minus 1.

For further information on the largest prime discovered to date, look in on http://www.utm.edu/research/primes/notes/1257787.html.

 

*  One way to determine whether a number N is prime is to divide it by all the primes less than sqrt(N). It would not be necessary to divide by primes greater than sqrt(N) as this would produce numbers that were already covered in the array of numbers less than srt(N). It would appear that this method is relatively simple, and so it is, for relatively small numbers. Large numbers however would require many tiresome calculations either by hand or with a calculator.

 

* Wilson's Theorem

   Wilson's Theorem stated that for every prime number "p", [(p + 1)! + 1] is evenly divisible by "p". The converse was shown to also be true in that every integer "n" that evenly divides [(n + 1)! + 1] is prime. Combining these leads to the famous general theorem that a necessary and sufficient condition that an integer "n" be prime is that "n" evenly divide [(n + 1)! + 1].

   Example: For N = 5, (5 - 1)! + 1 = 24 + 1 = 25 is divisible by 5. For N = 11, (11 - 1)! + 1 = 3,628,000 + 1 = 3,628,001 is divisible by 11. On the other hand, for N = 12, (12 - 1)! + 1 = 39,916,800 + 1 = 39,916,801 is not divisible by 12. 

 

*  Fermat's Theorem

   Fermat's Little Theorem stated that if "a" is any whole number and "p" is a prime number relatively prime to "a", then "p" divides [a^(p-1) - 1]. A corollary to this is if "p" is any prime number and "a" is any whole number, then "p" divides [a^p - a]. With p = 7 and a = 12, the theorem tells us that 7 divides [12^6 - 1] = 2,985,983 which it does, yielding 426,569. Similarly, 7 divides [12^7 - 12] = 35,831,796 yielding 5,118,828. Trying p = 8 and a = 12, we get [12^7 - 1]/7 = 4,478,975.875, confirming that 8 is not prime.

 

   As with Wilson's theorem, it was logically asked if the converse of Fermat's Theorem was also true, i.e., is every integer "n" that evenly divides [a^(p-1) - 1] or [2^n - 2] a prime? This was pursued by Chinese mathematicians in exploring if "n" divides [2^(n-1) - 1], is "n" prime? They initially concluded that "n" would be prime if it divided [2^(n-1) - 1] but were later proven wrong when it was discovered that while the number 341 did evenly divide [2^340 - 1], 341 was itself composite, being equal to 11 times 31. There are supposedly an infinite number of composite numbers, "n", both odd and even, that divide (2^n - 2). They are referred to as pseudoprimes. When it was discovered that there were composite numbers that divided all (a^n - a), they were called absolute pseudoprimes. The smallest of the absolute pseudoprimes is 561.  

 

*  The square root of any prime number is an irrational number.

 

*  Goldbach's Conjecture - Every even number greater than 3 is the sum of two prime numbers. No proof exists.

 

*   A complete list of prime numbers may be found at http://primes.utm.edu/lists/small/10000.txt.

 

*  A prime "p" is either relatively prime to a number "n" or divides it.

 

*  A product is divisible by a prime "p" only when "p" divides one of the factors.

 

*  A product q1xq2x.....qr of prime factors qi is divisible by a prime "p "only when "p" is equal to one of the qi's.

 

*  The number 5 is the only prime that is both the sum and difference of two primes.

 

*  All primes other than 2 are of the form 4x + 1 or 4x - 1.

 

*  All odd primes can be expressed as the difference of two squares in only one way.

   Set the two factors of the prime p equal to (x + y) and (x - y) and solve for x and y. Then, x^2 - y^2 = p.

   Example: Using the prime 37, (x + y) = 37 and (x - y) = 1. Adding, 2x = 39 or x = 19 making y = 18.

   Thus, 19^2 - 18^2 = 361 - 324 = 37.

 

*  Primes of the form 4x + 1 can be expressed as the sum of two squares in one way only.

   Therefore, any prime that exceeds a multiple of 4 by 1 is the hypotenuse of only one right triangle.

 

*  Primes of the form 4x - 1 cannot be expressed as the sum of two squares in any way.

 

*  Primes of the form Mp = 2^p - 1 are called Mersenne primes where the exponent "p" is itself a prime. Mersenne primes contribute to the derivation of perfect numbers. A perfect number is one that is equal to the sum of its aliquot divisors, i.e., all its divisors except the number itself. (It is also said that a number is perfect if the sum of all its divisors, including the number itself, is equal to twice the number.) Numbers of the form 2^(p-1)[2^p - 1] are perfect when (2^p - 1) is a prime. For example, the primes 2, 3, 5, 7, 13, etc., lead to the Mersenne primes of 3, 7, 31, 127, 8191, etc., and the corresponding perfect numbers of 6, 28, 496, 8128, 33,550,336, etc. There are no odd perfect numbers, or rather, none have been discovered to date.

 

*  The 48th Mersenne prime, 257,885,161 − 1 was found in 2013. It has more than 17 million digits.

 

*  When a product ac is divisible by a number b that is relatively prime to a, the factor c must be divisible by b.

 

*  When a number is relatively prime to each of several numbers, it is relatively prime to their product.

 

*  Pierre de Fermat once hypothesized that all numbers of the form F(n) = [2^(2^n)] + 1 were primes. This worked for n = 1, 2, 3, and 4 but Euler discovered in 1732 that [2^(2^5) + 1] = 4,294,967,297 = 6,700,416x641.

 

*  Marin Mersenne hypothesized that Mersenne numbers of the form Mp = 2^p - 1, p a prime number, were prime for a select group of primes. In 1947, an electronic computer showed that some were composite and discovered others that were prime. When "p" is not a prime, the Mersenne numbers are composite. Those that were confirmed as primes and all derived since are now referred to as Mersenne Primes. There are relatively few Mersenne Primes. One discovered in 1997 had a p = 2,976,221 with 895,932 digits. The 35th Mersenne Prime was discovered in late 1996, being 2^1,398,269 - 1 and having 420,921 digits.

The largest prime discovered as of June 1, 1999 is 2^6,972,593 - 1. Mersenne Primes contribute to the derivation of Perfect Numbers.

 

*  There are many formulas that yield primes but not all primes. The most misleading is x^2 - x + 41 which works fine up to x = 39 but breaks down for x = 40 and beyond. Others are 2x^2 - 199, x^2 + x + 11, x^2 + x + 17, x^2 - 79x + 1601, 6x^2 + 6x + 31, x^3 + x^2 - 349, to name a few. There are no practical formulaic ways to produce all the primes. The Mills formula conceptually is a proven formula but is not calcuable. Rowland's recurrence work in 2008 provides an ineffective way to produce only primes numbers (or 1).

 

*  There are an infinite number of Primes

Proof: 

From Euclid's Elements (Proposition 20, Book IX)

Assume that p1, p2, p3, ......pn is a finite set of prime numbers.

Create the product P = p1(p2)p3.......(pn) and add 1.

P + 1 forms a new number which is not divisible by any of the given set of primes and must therefore itself be a prime or it contains as a factor a prime differing from those already defined.

If P + 1 is prime, then it is clearly a new prime not of the original set of primes.

If P + 1 is not prime, it must be divisible by some prime q.

However, q cannot be identical to any of the given prime numbers, p1, p2, p3,......pn, as it would then divide both P and P + 1 and consequently, their difference = 1.

However, q must be at least 2 and cannot therefore divide evenly into 1.

Therefore, q, being different from all the given primes, must be a new prime.

 

Caution: Care must be taken to realize that p1(p2)p3.......(pn) + 1 will not always produce a new prime and if it does, it is not necessarily the next prime.

Examples:

2x3 + 1 = 7 = a prime

2x3x5 + 1 = 31 = a prime

2x3x5x + 1 = 211 = a prime

2x3x5x7x11 + 1 = 2311 = a prime

2x3x5x7x11x13 + 1 = 30,031 = 59x509

 

*  There are an infinite number of primes.

   Euclid's proof: Is there a prime greater than N? Derive the quantity N! + 1, N factorial plus one. (N! + 1) is therefore not divisible by any number up to, and including, N, as there is always a remainder of 1. If (N! + 1) is prime, it has no factors other than 1 and itself; if it is not prime, it has factors in addition to 1 and itself.

 

*  As the number of primes decrease and the distance between primes increases, the corresponding sets of consecutive composite numbers become longer. Within the first 100 numbers, there are seven composite numbers between 89 and 97. Within the numbers from 1 to 1000, there are 19 composite numbers between 887 and 907. To define N consecutive composite numbers, you need only compute the numbers (N + 1)! + 2, (N + 1)! + 3, (N + 1)! + 4, (N + 1)! + 5,..... .....(N + 1)! + (N + 1), all being composite.

 

*  Primes can be in arithmetic progression, for example:

   7-37-67-97-127-157, 107-137-167-197-227-257, 199-409-619-829-1039-1249-1459-1669-1879-2089.

 

*  There are an infinite number of palindromic primes such as 101, 131, 151, 181, 313, 353, 727, 757, 797, 919, 79,997, 91,019, 3,535,353, etc.

 

*  Primes can be both palindromic and in arithmetic progression such as in

   13931-14741-15551-16361, 10301-13331-16361-19391 and 94049-94349-94649-94949.

 

*  Unproven hypotheses concerning primes (only a few shown):

   Every even number => 4 is the sum of two primes.

   Every odd number => 9 is the sum of three odd primes.

   Every odd number > 4 can be expressed as a prime and the product of 2 consecutive integers eg n = p + x(x+1)

   Every even number is the difference between two primes in an infinite number of ways.

   Every number is the difference of two consecutive primes in an infinite number of ways.

   Every odd number >=7 is either a prime or the sum of a prime and twice a prime Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product. There are an infinite number of Twin Primes. A prime always exists between 2 consecutive perfect squares

   Primes can be expressed as the sum of a prime and twice a square.

  

                            (Remember - all unproven)

 

*  Goldbach's Conjecture - Every even number is the sum of two prime numbers. No proof exists

 

*  Any positive integer can be written as the product of primes in only one way.

      Example: 42 = 2x3x7, 67 = 1x67, 96 = 3x2^5, 52 = 13x2^2.

 

*  Using the digits 1 through 9 only once, form prime numbers the sum of which is a minimum.

   2 + 3 + 5 + 41 + 67 + 89 = 207.

   Perform the same thing using the digits 0 through 9.

   2 + 5 + 7 + 63 + 89 + 401 = 567.

 

*The squares of the first 7 primes, 2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2, add up to 666, the devil number.

 

While the above treatment of prime numbers appears long, it was considered worthwhile in that prime numbers are the root of all the remaining numbers other than prime numbers. Newer techniques include elliptical curves and Miller-Rabin method.

 

PRODUCT PERFECT

 

We call a number a product perfect number if the product of all its divisors, other than itself, is equal to the number. For example, 10 and 21 are product perfect numbers since 1*2*5 = 10 and 1*3*7 = 21, whereas 25 is not, since the product of its divisors, 1*5 = 5 is too small. I have listed all the product perfect numbers between 2-60, which are: 6,8,10,14,15,21,22,26,27,28,33,34,35,38,39,44,45,46,51,52,55,57,58. I am stumped on the description - a description that should make it easy for you to produce a six-digit product perfect number with very little work.

 

PRONIC NUMBERS

 

Pronic numbers are the product of two consecutive numbers. They are synonymous with oblong numbers, numbers that represent the total number of dots contained within a rectangular arrangements of dots with the number of columns being one more than the number of rows. Coincidentally, the oblong numbers are also the sums of the first "n" even integers as defined by the expression Pn = n(n + 1). They are also twice the triangular numbers since Tn = n(n + 1)/2.

 

PSEUDOPRIME NUMBERS

 

One of Pierre de Fermat's famous theorems stated that if "p" is a prime number and "a" is a number smaller than "p" and relatively prime to "p", then [a^(p-1) - 1] or [a^p - a] will be evenly divisible by "p". This was proven by Euler and others. This led to a Chinese theorem that explored the converse interpretation that if [2^(p-1) - 1] or 2^p - 2 is evenly divisible by "p", then "p" is a prime number and if  not evenly divisible by "p", then "p" is not prime. While initially accepted as being true, it was ultimately discovered that if, in fact, [2^(p-1) - 1] is evenly divisible by "p", it does not necessarily mean that "p" is a prime as numbers were discovered that satisfied the theorem's requirement, yet were not prime. It was first discovered that [2^90 - 1] was evenly divisible by 91 and [2^340 - 1] was evenly divisible by 341 but 91 = 7x13 and 341 = 11x31. It was then that composite numbers "p" that did evenly divide [2^(p-1) - 1] were called pseudoprimes. They were ultimately called Poulet numbers after  a mathematician who derived and published a list of odd pseudoprimes up to 100 million.

 

With only 4 numbers out of the first 1000 being pseudoprimes, 91, 341, 561, and 645, Fermat's Theorem only tells us that if [a^(p-1) - 1] is divisible by "p", "p" can be a prime or a pseudoprime. It is widely believed that there are an infinite number of pseudoprimes. The term absolute pseudoprime was ultimately given to those composite numbers "a" that divided all [2^(p-1) - 1], the number 561 being the smallest.

 

There are Poulet numbers where all the factors/divisors divide [2^(p-1) - 1] and are called super-Poulet numbers.

 

PYRAMIDAL NUMBERS

 

Pyramidal numbers are the three dimensional representations of polygonal numbers. Stacking triangular, square, pentagonal, etc., numbers results in the pyramidal numbers.

 

The following is a depiction of a three dimensional triangular pyramid as viewed from one side of the pyramid, the third dimension, in and out of the paper, being of the same size and shape as the in plane shape and size shown, i.e., each successive vertical layer is the next triangular number.

 

Triangular Pyramids or Tetrahedral Numbers

Rank.........1........2...............3.....................4.............................5.....................6.......7........8........9

Tn.....1......O.......O...............O....................O............................O

.........3.............O...O.........O...O..............O...O......................O...O

.........6............................O...O...O.......O....O....O..............O....O....O

........10..............................................O....O...O....O.......O.....O...O....O

........15........................................................................O....O....O....O....O

Triangular Pyramidal/Tetrahedral Numbers

................1.......4................10...................20............................35..................56.....84.....120.....201.....etc.

 

   The nth tetrahedral number is given by PyT(n) = n(n + 1)n + 2)/6.

 

Square Pyramids

   The following is a depiction of a three dimensional square pyramid as viewed from one side of the pyramid, the third dimension, in and out of the paper, being square and of the same size as the in plane dimension shown.

 

Rank.........1........2...............3.....................4.............................5..................6........7........8.........9

Tn...1........O.......O...............O....................O............................O

......4................O...O.........O...O..............O...O......................O...O

......9...............................O...O...O.......O....O....O..............O....O....O

.....16.................................................O....O...O....O.......O.....O...O....O

.....25............................................................................O....O....O....O....O

Square Pyramidal Numbers

................1.........5..............14....................30...........................55...............91....140.....204.....285..etc.

 

   The nth square pyramidal number is given by PyS(n) = n(n + 1)2n + 1)/6.

 

   Pyramids of triangular and square bases are the only two pyramids that can be built with nested spheres. All other polygonal numbers cannot uniformly fill all of the spaces between adjacent spheres.

 

Further pyramidal numbers derive from:

Triangle - n(n + 1)(n + 2)/6

Square - n(n + 1)(2n + 1)/6

Pentagon - n^2(n _ 1)/2

Hexagon - n(n + 1)(4n - 1)/6

Heptagon - n(n + 1)(5n - 2)/6

Octagon - n(n + 1)(2n - 1)/2

 

Pyramidal numbers are the three dimensional representations of polygonal numbers. Stacking triangular, square, pentagonal, etc., numbers results in the pyramidal numbers.

 

Triangular numbers - .......1, 3,   6, 10, 15, 21, 28,  36,   45,   55, etc. = n(n + 1)/2

Triangular pyramidals -.....1, 4, 10, 20, 35, 56, 84, 120, 165, 220, etc. = n(n + 1)(n + 2)/6

Square numbers -............1, 4,   9, 16, 25, 36,  49,   64,   81,  100, etc. = n^2

Square pyramidals -.........1, 5, 14, 30, 55, 91, 140, 204, 285, 385, etc. = n(n + 1)(2n + 1)/6

Pentagonal numbers -......1, 5, 12, 22, 35,  51,   70,   92, 117, 145, etc. =

Pentagonal pyramidals -...1, 6, 18, 40, 75, 126, 196, 288, 405, 550, etc. =  


Continue This Article | Part I | Part II | Part III

 

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