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^(p1)(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 semiperfect
number is a number that is equal to the sum of some of its
aliquot parts or proper divisors. The smallest semiperfect
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
semiperfect, multiplyperfect, quasiperfect, 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
1234.....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(3n1)/2
Hexagonal
numbers, Hxn, derive from 1 + 5 + 9 + 13 + 17 + 21 + .......= 1,
6, 15, 28, 45, 66, 91.....= n(2n1)
Heptagonal
numbers, Hpn, derive from 1 + 6 + 11 + 16 + 21 + 26 + ... = 1,
7, 18, 34, 55, 81, 112.....= n(5n3)/2
Octagonal
numbers, On, derive from 1 + 7 + 13 + 19 + 25 + 31 + ...... = 1,
8, 21, 40, 65, 96, 133.....= n(3n2)
ngonal =
n[(n1)n  2(n2)]/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(r1) where Sr is the square number of rank r and
Tr is the triangular number of rank r. Since Tr = r(r1)/2 and
T(r1) = (r1)(r1+1)/2 = (r1)r/2, we can write Tr + T(r1) =
r(r+1+r1)/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(n1). For
instance, the 4th pentagonal number P4 = S4 + T3 = 16 + 6 =
22. Similarly, Hxn = Sn + T(n2).
Similarly, Hxn
= Pn + T(n1), Hpn = Hxn + T(n1), On = Hpn + T(n1), 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(n1))^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:
1Recreations
in the Theory of Numbers by Albert H. Beiler, Dover
Publications, Inc., 1964.
2Mathematical Recreations and Essays by W.W. Rouse and H.S.M.
Coxeter, Dover Publications, Inc., 1987.
3The Book of
NUMBERS by J.H. Conway and R.K. Guy, Copernicus/SpringerVerlag,
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
1a positive
number "p" that has but two positive divisors/factors, 1 and "p"
or
2any 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 nondecreasing 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
semiprime 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 semiprime
numbers. By definition, such numbers are also classified as
composite numbers.
* Primes
differing by 2 are called twin primes.
Examples:
35, 57, 1113, 1719, 2931, 4143, 5961, 7173, 101103,
1,000,000,009,6491,000,000,009,651.
* The sums of
all pairs of twin primes (except 35) 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^(p1)  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^(p1)  1] or [2^n  2] a prime? This was
pursued by Chinese mathematicians in exploring if "n" divides
[2^(n1)  1], is "n" prime? They initially concluded that "n"
would be prime if it divided [2^(n1)  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^(p1)[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:
7376797127157, 107137167197227257,
199409619829103912491459166918792089.
* 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
13931147411555116361, 10301133311636119391 and
94049943499464994949.
*
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 MillerRabin 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 260, 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 sixdigit 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^(p1)  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^(p1)  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^(p1)  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^(p1)  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^(p1)  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^(p1)  1],
the number 561 being the smallest.
There are
Poulet numbers where all the factors/divisors divide [2^(p1) 
1] and are called superPoulet 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.
=
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