Download Perfect Numbers and Mersenne Primes

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Proper Divisor: The proper divisors of a number are all its divisors
(factors) excluding the number itself.
Taking 36 as an example:
Its proper divisors are 1, 2, 3, 4, 6, 9, 1 2, and 18 but not 36.
In the investigation that follows we will only consider proper divisors.
Perfect Numbers
Abundant Numbers
Deficient Numbers
Mersenne Primes
Weird Numbers
Factors:
12
1, 2, 3, 4, 6, 12
18
1, 2, 3, 6, 9, 18
15
1 + 2 + 3 + 4 + 6 = 16
16 > 12
Abundant Number
1 + 2 + 3 + 6 + 9 = 21
Abundant Number
21 > 18
1, 3, 5, 15
1+3+5=9
9 < 15
Deficient Number
12
18
15
Abundant
Abundant
Deficient
6
1, 2, 3, 6
P1 = 6
1+2+3=6
6 = 6  Perfect Number
Perfect Number
Check the factors of the numbers on your list to see if
they are Abundant, Deficient or Perfect. Can you find
P2, the second perfect number?
Factors
A/D/P
Factors
1
25
2
26
3
27
4
28
5
29
6
1, 2, 3
P
30
7
31
8
32
9
33
10
34
11
35
12
1, 2, 3, 4, 6
A
36
13
37
14
38
15
1, 3, 5
D
39
16
40
17
41
18
1, 2, 3, 6, 9
A
42
19
43
20
44
21
45
22
46
23
47
24
48
A/D/P
Also consider:
1. The distribution of abundant
and deficient numbers.
2. Numbers with the fewest
factors.
3. Numbers with the most
factors.
There are more Factors
deficient numbersA/D/P
1
than
abundant numbers and all the
2
abundant
numbers are even.
3
1, 5
D
26
1, 2, 13
D
27
1, 3, 9
D
28
1, 2, 4, 7, 14
P
29
1
30
1, 2, 3, 5, 6, 10, 15
31
1
32
1, 2, 4, 8, 16
D
33
1, 3, 11
D
34
1, 2, 17
D
35
1, 5, 7
D
A
36
1, 2, 3, 4, 6, 9, 12, 18
A
Prime
37
1
4
1
A/D/P
25
Obviously
the prime numbers have
5
only
one
factor.
6
1, 2, 3
P
7
Factors
Prime
Once
you have done the “Product of
8
1, 2, 4
D
Primes”
9
1, 3 presentation you may be D
able
to
why numbers such as 24,
10
1, 2,see
5
D
11 40,
1 and 48 have lots of factors.
Prime
36,
Prime
A
Prime
12
1, 2, 3, 4, 6
13
1
14
1, 2, 7
D
38
1, 2, 19
D
15
1, 3, 5
D
39
1, 3, 13
D
16
1, 2, 4, 8
D
40
1, 2, 4, 5, 8, 10, 20
A
17
1
Prime
41
1
18
1, 2, 3, 6, 9
A
42
1, 2, 3, 6, 7, 14, 21
19
1
Prime
43
1
20
1, 2, 4, 5, 10
A
44
1, 2, 4, 11, 22
D
21
1, 3, 7
D
45
1, 3, 5, 9, 15
D
22
1, 2, 11
D
46
1, 2, 23
D
23
1
Prime
47
1
24
1, 2, 3, 4, 6, 8, 12
A
48
1, 2, 3, 4, 6, 8, 12, 16, 24
Prime
Prime
A
Prime
Prime
A
For Homework:Factors
A/D/P
1
1, 2, 4
A/D/P
25
1, 5
D
26
1, 2, 13
D
27
1, 3, 9
D
28
1, 2, 4, 7, 14
P
29
1
30
1, 2, 3, 5, 6, 10, 15
31
1
32
1, 2, 4, 8, 16
D
33
1, 3, 11
D
34
1, 2, 17
D
35
1, 5, 7
D
36
1, 2, 3, 4, 6, 9, 12, 18
A
Prime
37
1
2
There
is only one weird number
3
below
100 can you find it? A weird
4
number is an abundant number that
5
cannot be written as the sum of any
6
1, 2, 3
P
subset
of
its
divisors.
7
1
Prime
8
Factors
D
As
20 is not weird since
9 an1,example:
3
D it
can
written
as 1 + 4 + 5 + 10 andD
10 be
1, 2,
5
11 is not
1
Prime
36
weird since it can be written
3, 4,+6 6 or 9 + 6 + 18 + 3
A
as12 18 1,+2,12
Prime
A
Prime
13
1
Prime
14
1, 2, 7
D
38
1, 2, 19
D
15
1, 3, 5
D
39
1, 3, 13
D
16
1, 2, 4, 8
D
40
1, 2, 4, 5, 8, 10, 20
A
17
1
Prime
41
1
18
1, 2, 3, 6, 9
A
42
1, 2, 3, 6, 7, 14, 21
19
1
Prime
43
1
20
1, 2, 4, 5, 10
A
44
1, 2, 4, 11, 22
D
21
1, 3, 7
D
45
1, 3, 5, 9, 15
D
22
1, 2, 11
D
46
1, 2, 23
D
23
1
Prime
47
1
24
1, 2, 3, 4, 6, 8, 12
A
48
1, 2, 3, 4, 6, 8, 12, 16, 24
Prime
A
Prime
Prime
A
THE SCHOOL of ATHENS (Raphael) 1510 -11
Socrates
Pythagoras
Plato
Aristotle
Euclid
“All Men by nature desire knowledge”: Aristotle.
The Mathematicians of Ancient Greece.
Pythagoras
of Samos
(570 – 500 BC.)
P1 = 6
P2 = 28
P3 = 496
P4 = 8128
Euclid of
Alexandria
(325 – 265 BC.)
Archimedes
of Syracuse
(287 – 212 BC.)
Eratosthenes
of Cyene
(275-192 BC.)
1+2
+ 3 mathematicians
=6
The
of Ancient Greece
knew the first 4 perfect numbers and
1 + 2 + 4 + 7 + 14 = 28
the search was on for the P5
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016
+ 2032 + 4064 = 8128
The Mathematicians of Ancient Greece.
Pythagoras
of Samos
(570 – 500 BC.)
P1 = 6
P2 = 28
P3 = 496
P4 = 8128
Euclid of
Alexandria
(325 – 265 BC.)
Archimedes
of Syracuse
(287 – 212 BC.)
Eratosthenes
of Cyene
(275-192 BC.)
How many digits would you reasonably expect P5
to have and what is the largest number that you
can make with this many digits?
Five digits seems reasonable considering the
digit sequence 1,2,3,4… and 99,999 is the
highest 5 digit number.
The Mathematicians of Ancient Greece.
Pythagoras
of Samos
(570 – 500 BC.)
P1 = 6
P2 = 28
P3 = 496
P4 = 8128
Euclid of
Alexandria
(325 – 265 BC.)
Archimedes
of Syracuse
(287 – 212 BC.)
Eratosthenes
of Cyene
(275-192 BC.)
The Greeks never managed to find it. This elusive
number turned up in medieval Europe in an
anonymous manuscript and it was an 8 digit number
P5 = 33 550 336
P1= 6
P2= 28
P3= 496 P4 = 8128 P5 = ? (a 5 digit number?)
P5 = 33 550 336 (1456 Not Known) 8 digits
P6 = 8 589 869 056 (1588 Cataldi) 10 digits
P7 = 137 438 691 328 (1588 Cataldi) 12 digits
P8 = 2 305 843 008 139 952 128 (1772 Euler) 19 digits
P9 = 2 658 455 991 569 831 744 654 692 615 953 842 176 (1883 Pervushin)
37 digits
P10 = 191 561 942 608 236 107 294 793 378 084 303 638 130 997 321 548
169 216 (1911: Powers) 54 digits
P11 =13 164 036 458 569 648 337 239 753 460 458 722 910 223 472 318
386 943 117 783 728 128 (1914 Powers) 65 digits
P12 =14 474 011 154 664 524 427 946 373 126 085 988 481 573 677 491
474 835 889 066 354 349 131 199 152 128 (1876 Edouard Lucas) 77 digits
P13=23562723457267347065789548996709904988477547858392600710
143020528925780432155433824984287771524270103944969186640286
44534175975063372831786222397303655396026005613602555664625
032701752803383143979023683862403317143592235664321970310172
071316352748729874740064780193958716593640108741937564905791
8549492160555646 976 (1952 Robinson) 314 digits
As you can see perfect numbers are extremely rare.
There are only 41 known perfect numbers. The largest
P41 (discovered in 2004) has 14 471 465 digits.
There is a strong link between perfect numbers and other
numbers called Mersenne Primes. Mersenne primes are as
rare as perfect numbers, M41 is the largest known prime just
as P41 is the largest known perfect number. In fact every
time someone finds a new one they can calculate the resulting
perfect number by use of a formula. For every Mersenne
Prime there is a corresponding Perfect number. The 41st
Mersenne Prime has 7 235 733 digits. Each perfect number
has roughly double the number of digits as its corresponding
Mersenne prime. Each time one is discovered it automatically
becomes the world’s largest known prime.
There is a $100,000 prize for the first person that
discovers a prime with over 10 million digits and it could
be you!
Monday 7th June 2004
Largest Prime Discovered!
A scientist has used his computer to find the largest prime
number found so far.- written out, it would stretch for 25
km. The number is the Mersenne prime 224 036 583 –1. The
new figure identified by Josh Findley contains 7,235,733
digits and would take someone the best part of 6 weeks to
write out by hand. Mr Findley was taking part in a mass
computer project known as GIMPS (Great Internet
Mersenne Prime Search). Gimps is closing in on the $100
000 prize for the first person to find a 10-million–digitprime! “An award winning prime could be mere weeks away
or as much as a few years away” said GIMPS founder
George Woltman.
41st Mersenne
Prime Found!
• Primes are the
building blocks of all
whole numbers.
Historically searching for Mersenne primes has been used to test computer hardware.
The free GIMPS programme used by Findley has identified hidden hardware problems in
many computers. He used a 2.4 GHz Pentium 4 Windows XP computer running for 14
days to prove the number was prime. Prime numbers are mainly of interest to
mathematicians that study the branch of mathematics called Number Theory but they
are becoming important in cryptography and may eventually lead to uncrackable codes.
Mersenne Primes
A Mersenne number is any number of the form 2n – 1.
The first 12 of these are shown below.
21 – 1 = 1
22 – 1 =
3
26 – 1 = 63
210 – 1 = 1023
23 – 1 =
7
27 – 1 = 127
24 – 1 =
15
28 – 1 = 255
211 – 1 = 2047
25 – 1 =
31
29 – 1 = 511
212 – 1 = 4095
Early mathematicians thought that if n was prime then 2n -1 was also prime.
Mersenne Primes
A Mersenne number is any number of the form 2n – 1.
The first 12 of these are shown below.
21 – 1 = 1
22 – 1 =
3
26 – 1 = 63
210 – 1 = 1023
23 – 1 =
7
27 – 1 = 127
24 – 1 =
15
28 – 1 = 255
211 – 1 = 2047
25 – 1 =
31
29 – 1 = 511
212 – 1 = 4095
Early mathematicians thought that if n was prime then 2n -1 was also prime.
In 1536 Hudalricus Regius showed that 211 -1 = 23 x 89 and so was not prime.
Mersenne Primes
A Mersenne Prime is any prime number of the form 2n – 1.
were n is prime.
22 – 1 =
3
23 – 1 =
27 – 1 = 127
25 – 1 =
7
31
A French monk called Marin Mersenne stated in
one of his books in 1644 that for the primes:
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257
1588 - 1644
that 2n -1 would be prime and that all other
positive integers less than 257 would yield only
composite (non-prime) numbers. He stated this
even though he could not possibly have checked
such huge numbers. He was making a conjecture.
After this all such numbers that were prime
became known as Mersenne Primes. In
subsequent years various mathematicians showed
that his conjecture was not correct.
Mersenne Primes
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
27 – 1 = 127
261 – 1
289 – 1
213 – 1
2107 – 1
217 – 1
2127 – 1
Mersenne’s List
n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257
1588 - 1644
Two of the numbers on Mersenne’s list did not
generate Mersenne primes and there were
others that he had missed off.
Completed List
n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127,
It was 1947 before all the numbers on Mersenne’s original list had been checked
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
213 – 1
27 – 1 = 127
261 – 1
289 – 1
2107 – 1
217 – 1
2127 – 1
There is a formula linking a Mersenne Prime to its
corresponding perfect number by multiplication.
Use the table below to help you find it.
1588 - 1644
Perfect
Number
n
2n -1
2
3
x
?
6
3
7
x
?
28
5
31
x
?
496
7
127
x
?
8128
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
213 – 1
27 – 1 = 127
261 – 1
289 – 1
2107 – 1
217 – 1
2127 – 1
There is a formula linking a Mersenne Prime to its
corresponding perfect number by multiplication.
Use the table below to help you find it.
1588 - 1644
Perfect Write as a
Number power of 2
n
2n -1
2
3
x
2
6
3
7
x
4
28
5
31
x
16
496
7
127
x
64
8128
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
213 – 1
27 – 1 = 127
261 – 1
289 – 1
2107 – 1
217 – 1
2127 – 1
There is a formula linking a Mersenne Prime to its
corresponding perfect number by multiplication.
Use the table below to help you find it.
1588 - 1644
Perfect Write as a
Number power of 2
n
2n -1
2
3
x
2
6
21
3
7
x
4
28
22
5
31
x
16
496
24
7
127
x
64
8128
26
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
213 – 1
27 – 1 = 127
261 – 1
289 – 1
2107 – 1
217 – 1
2127 – 1
If 2n -1 is a Mersenne prime then 2n – 1 x 2n-1 is a
perfect number. Check this for the first few.
1588 - 1644
Perfect Write as a
Number power of 2
n
2n -1
2
3
x
2
6
21
3
7
x
4
28
22
5
31
x
16
496
24
7
127
x
64
8128
26
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
27 – 1 = 127
261 – 1
213 – 1
289 – 1
2107 – 1
217 – 1
2127 – 1
If 2n -1 is a Mersenne prime then 2n – 1 x 2n-1 is a
perfect number. Check this for the first few.
2 2 – 1 x 21 = 3 x 2 = 6
23 – 1 x 22 = 7 x 4 = 28
1588 - 1644
25 – 1 x 24 = 31 x 16 = 496
27 – 1 x 26 = 127 x 64 = 8128
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
27 – 1 = 127
261 – 1
289 – 1
213 – 1
2107 – 1
217 – 1
2127 – 1
NEWS FLASH
4th September 2006
44th Mersenne Prime Found.
232 582 657 – 1 has 9,808,358 digits
1588 - 1644
The $100 000 prize for the world’s
first 10 million digit prime is still on
but you need to be quick.
Mersenne Primes and Perfect Numbers
A Mersenne Prime is any number of the form 2n – 1. were n
is prime and produces a prime number..
22 – 1 =
3
23 – 1 =
7
219
25 – 1 =
31
– 1 231 – 1
27 – 1 = 127
261 – 1
289 – 1
213 – 1
2107 – 1
217 – 1
2127 – 1
Research other information about
Mersenne Primes and Perfect Numbers
and don’t forget to join GIMPS.
1588 - 1644
http://www.mersenne.org/
Great Internet Mersenne Prime Search
Factors
A/D/P
Factors
1
25
2
26
3
4
Worksheet 1
5
29
6
30
7
31
8
32
9
33
10
34
11
35
12
36
13
37
14
38
15
39
16
40
17
41
18
42
19
43
20
44
21
45
22
46
23
47
24
48
27
28
A/D/P
Perfect
Number
n
2n -1
2
3
x
6
3
7
x
28
5
31
x
496
7
127
x
8128
n
2n -1
2
3
x
6
3
7
x
28
5
31
x
496
7
127
x
8128
Perfect
Number
Worksheet 2