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Matt: Thank you.
Matt: OK. Let's Get started. I'm Matt
Stath. In October, Krishna sent an email to
The Math Club about arithmetic progressions
in square numbers. I mean I emailed him
back about arithmetic progressions in cal-
culating primes. I had discovered some pro-
bable primes with 10,000 or more digits in
arithmetic progressions and he invited me
to speak. First I'll have some videos.
TheSingingNerd on Youtube
Matt: This is the largest
known prime number.
Gatsby428 on Youtube
Matt: The 17,425,170 digits divided by
3,486 digits per page is 4,998 2/3
pages. Printed on both sides of the paper
it would be 2,499 1/3 pages or 5/6 of
this box of computer printer paper. Here's
the second largest known prime.
AssociatedPress on Youtube
Matt:...and it's estimated that the number
of subatomic particles in the universe is
10^80. So, it would be more than the sub-
atomic particles in the universe. Almost
12,978,189 digits divided by the number of
digits per page would be more than 3,723
pages. Printed on both of the paper, it
would be 1,861 1/2 pages or 5/8 of a
box of printer paper. These use GIMPS, The
Great Internet Mersenne Prime Search. In
case you were wondering from the previous
video, the first 100 million digit base 2
number would be this. But it's not very
likely to be prime. That one would be
4 4/5 boxes of printer paper. Mersenne
primes are 2^(prime number)-1. I'll talk
about origins of primes. The Ishango
bones of baboons discovered in 1960 in
Belgian Congo from about 20,000 BC or
Lebombo bones of Swaziland in 3,500 BC.
Amitavi: 35,000.
Matt: What!?
Amitavi: 35,000.
Matt: ...and if somebody thinks that I'm
going too fast then they can just tell me
to move back a slide or you can ask about
something. Let's take a closer look at The
Ishango Bones. They have odd numbers.
There are different theories about what they
mean and there's a debate. One theory is
that they use Egyptian Math, Base 60 Math,
that they are cycles of The Moon, or women
keeping track of their periods. The Sumer-
ian Cuneiform in ancient Iraq had a list
of primes according to Vern Foley, one of
my History professors. They were base 60.
So it would be 2, 3, 5, 7, 11, and so on.
One theory about why The Sumerians used
base 60 was that they had problems with
fractions, or that there are 360 degrees
in a circle, or about 360 days in a year.
Let's talk about Titanics, numbers with
1,000 or more digits. Archimedes was fa-
mous for figuring out how much of a king's
crown was gold. He's famous for The Arch-
imedes spiral among other things. His
"Myriad of Myriads" is
(100 million)^(100 million) or 1 followed
by 80 million billion zeros. It obviously
isn't prime since it ends with a zero. He
believed in The Sun god or Sun Titan Hy-
perion and had a riddle about how many
cattle were needed to pull the Sun across
the sky. It was 5,916,837,175,686 or a
multiple of a 206,545-digit number. A
206,545-digit number? Well, before you
jump to the conclusion that he was on
mushrooms, it may have been propaganda
against Archimedes to make him look crazy
because there was a war, or they wanted
his land, or they thought he was a threat
to the Roman empire. "The Mathematics
Teacher" newsletter in 1971 published a
1,560-digit number, a bracelet number,
but it wasn't prime. The same publication
mentioned 1 with 1,000 zeros in 1973.
Samuel Yates defined Titanic primes as
1,000 or more digits. Gigantic primes are
10,000 or more digits. Yates died and
Chris Caldwell took over the Top 5,000
database in Tennessee. It's a database
where you don't need a password. It's
open to the public. Let's just click on
search. Click on all. Let's find all of
the primes with a million or more digits.
There are 63. Somebody this month just
discovered 1 with exactly 1 million
digits. These are the primes from The
Youtube videos. There are also believed
to be 10^2,400,000,000 DNA combinations
in LIFE Science Library MATHEMATICS in
1972. It's interesting to read Math books
from before the internet. I thought it
was 4^(6 billion) power in humans because
there 4 combinations of base pairs. There
are 6 billion of them. I was in the 99
percentile in Math, I heard about the
number googol from elementary school, and
a Math show called Square One interested
me. Disney's Chip 'N' Dale Rescue Rangers
had a joke. The gazillionaire's wife had
so much money that she spent 3 weeks
writing the zeros on the check. I was in
3rd grade and I thought that's the
stupidest thing that they said on the
show all year! Because if she wrote down
a zero one second, 60 seconds * 60
minutes * 24 * 7 * 3, that would be more
than a million zeros. A million would
have 6 zeros, not a million zeros. Also,
I was in MathCounts. In 2003-04 reading
in The Purdue Exponent, I saw 2 computer
articles that caught my eye: A Leukemia
cure crowd sourcing screensaver that's
probably a precursor to Fold-It Beta.
They take different proteins and they
compute how they unfold in water or
enzymes. I just wasn't familiar with the
chemistry of radiation or chemotherapy,
so I didn't stick with it. But at about
the same time in The Purdue Exponent,
Purdue students tried to find an
equation to find a prime with over
40,000 digits and it didn't work. I don't
have that article anymore. I'm not sure
if their names were Chen and *** because
in the Little Book of Bigger Primes, Chen
and *** tried to find an equation with
over 40,000 digits and it did work. Also,
there's a 2005 Discover magazine article
about arithmetic progressions in primes.
You start with 199 and keep on adding
210. It's a 10-term arithmetic pro-
gression to find prime numbers. At that
time, the largest known number of primes
in an arithmetic progression was 22. You
start with 11,410,337,850,553. Go up in
steps of 4,609,098,694,200. Then there's
another one. Start with
376,859,931,192,959 and you go up in
steps of 18,549,279,769,020. I've
already showed you The Top 5,000 Primes
web page. About the year that I had read
the Discover magazine article, the
5,000th place prime had 70,000 digits.
Now the new prime in 5,000th place has
280,000 digits. There's also The Top
20's Complete Index with the different
themes of primes. These are just the
ones they care to list and they got
really creative with the Prime Curios.
There's also the Lifchitz brothers'
Probable Primes with 10,000 or more
digits. I'm now on here. I know, I email
Giovanni and Maksym from The Repunit
Project, who are also on here. I know
of Makoto Komado, who worked with
repunits. I'll explain the Repunit Pro-
ject in a little bit. Why are primes
important? There's public key
cryptography. You have Prime1 * Prime2
and it equals a composite. The com-
posite is a public key. With P1 and P2,
they are private keys. You can't guess
the prime number for a private key
because it would take over 1,000
years. Primes are also important for
education, hypotheses, Autistic
Savants like Daniel Tammet, Ulam's
Spiral - This is a Ulam's spiral, it
starts at 2, 3, 5, 7, and there's a dot
at every prime number, - theorems,
formulas, The Search for Extra-
terrestrial Intelligence - You might
like the movie, "Contact." They inter-
cept primes from space and primes just
don't happen in nature. If you receive
a signal that's just all the primes in
order, it has to come from Earth or
aliens and that's the premise of the
movie, "Contact." - Mersenne primes,
bee and rabbit family trees with
Fibonacci numbers, contests, prizes,
and etc. With Mersenne family trees,
we have 16 second-degree great
grandparents + 8 great grandparents
+ 4 grandparents, plus 2 parents, plus
yourself and that's 31, a prime number.
If you imagine that with 2 to the 57
millionth or 59 millionth, it goes
back very far. Rabbits have a gesta-
tion of a month, give birth, and are
1, 1, 2, 3, 5, 8, or 13 pairs in the
previous month. This is The Fibonacci
sequence. 1, 1, 2, 3, 5, 8, and 13. 13
is a Fibonacci number. With Pascal's
Triangle you go diagonally and it's a
Fibonacci number. Bees must have some-
thing strange with their chromosomes.
The male has 1 female parent. The
female has 2 parents, 1 female and 1
male. So it goes 1, 1, 2, 3, 5, 8, 13,
and so on. The largest probable Fibon-
acci prime is 411,439 digits and
discovered by one of the Lifchitz in
France. You'd have to email them if
you want to discover a Lifchitz, I
mean, a Fibonacci number. They have
the program. The Sacred Geometry, or
Golden Ratio, or Phi, 1.618033 is in a
lot of things in nature like nautilus
shells. I'll put variable n or x in
red in these slides. If anybody has a
question, then just interrupt, and ask
me. From Wolfram, the Euler formula is
x^2+x+41 and that's prime for x=0 to
39. I've discussed the Discover equa-
tions. 210*n+199 for n=0 to 9 and other
Discover magazine equations. So, the
first time that I discovered a prime
that I care to remember, I was trying
to figure out a 1,000 digit prime. So,
I took n in the equation n^2+n+41 and I
changed it to (10^499) and it happened
to be prime. But it was 999 digits,
not 1,000, so I was a little bit upset.
For now, I'm just happy that the compu-
ters are working 5 times faster than in
the year that I discovered it. That
might make you happy when you're trying
to find a prime. So I submitted it to
the Prime Curios page. I have these
PrimeForm equations. PrimeForm is a
program that they don't use anymore. I
have (10^n)^2, and I started at 0
digits, and as the n value went up, I
mean the x value, went up higher and
higher like it was on steroids you just
discover more and more primes with
10,000 or more digits. It took days to
discover primes. Sometimes I would
discover none in a night. Sometimes I
would discover 1 or 2. Here's another
one. 20,731 primorial. Now that pound
sign. That means that it's primorial.
It's all the prime numbers multiplied
together all the way up to that number.
But I couldn't include 41 because it's
over here. So I had ((n*2+1)#/41)^2 and
that's how that was calculated. I have
base 2 formulas and base 10 formulas.
You can just leave it running on your
computer when you go to class, or to
work, or to sleep. Other equations I've
discovered. But these are different,
these are with the equation, I mean the
formula, x^2-x+41. These are discovered
with a newer program, Open PFGW. Based
on that equation, I divided it by 199
and 11 so I pumped up n on steroids
again and I added 199. These are the
equations from Discover magazine that
are in the trillions. This is all on-
line. Anybody have any questions so
far? ...and here is another number
that's primorial. Generalized Fermat
numbers usually are
k*(an even base)^(of a power of 2)+1.
Now k can be equal to 1. So you have it
to the power of 2, 4, 8, 16, 32, and so
on. Somebody discovered one last year
with an exponent of more than half a
million and 2,976,633 digits. I'm
trying to find a Top 5,000 Generalized
Fermat in NewPGen, that's a sieve, with
over 410,000 digits. So I tried to alter
the equation, instead of an even number
for n, what if you had an odd number?
Then take it to the power of 4096 and
then you add 2. It works with 2048, 1024,
512, 256, and 128. Do you have a
question?
Kilian: Yes. Any particular reason why
you chose these numbers?
Matt: Oh, the Generalized Fermat numbers
are the most common numbers on the Top
5,000 list. If I noticed that if they
are that easy to discover I might as
well take an odd number, take it to a
power of 2, and then add 2. The reason
why those numbers are strange is that
it could quickly calculate which ones
would be the smallest 10,000 digit pro-
bable prime numbers.
That does not have more
than 100 digits, I mean that does not
have more than 100 characters. Henry and
Renaud will allow an equation with up to
100 characters on their web site.
...and with the different themes, a
palindrome is a number or phrase like,
"Ana, nab a banana."
Or "A man, a plan, a canal, panama."
"A man, a plan, a canal, panama."
"Ana, nab a banana." That reads the same
forwards and backwards. I'll have a
demonstration of this for a prime. This
is the most complicated equation I have
up here. But I was just playing around,
copying and pasting, copying
and pasting, and I had a
palindrome that's 271 digits. I worked
out what the equation would be. Here's an
n value where if you pump it up on ster-
oids if it gets changed to over 1,000
then you have a probable prime palindrome
with 23,611 digits. ...and this is
kind-of crazy. In the Commander Keen com-
puter games and home-brewed computer
games, people added a Dopefish as an
Easter egg just for the heck of it. These
are different digits for the border. It
took me a day to copy, paste that, and
it took me overnight for it to find the
value that's added to this to make it a
probable prime. Now it would be faster
because they have Open PFGW instead of
PrimeForm. If you think that you discover
a near-random prime, or a picture prime,
or primes that are over 100 characters
then since they are all over 100 char-
acters then you can have them linked to
me off of Mattstath.com. Let me demon-
strate how to find near-random primes.
This is a goofy name for a web site,
psychicscience.com where they have a good
random number generator. Generate 27,000
random integers between 0 and 9, open
sequence, and then it calculates them.
But It's only going to take only about a
minute to find a near-random digit number.
What I mean by that is I took a number
that I copied and pasted... Hold on, let
me find Microsoft Word. Where did they
move Microsoft Word in Windows 7?
Students: Go Backward. Start. Programs.
Yeah, just type in Word. Yeah, type in
Word. From the top.
Matt: OK. So it should take a couple min-
utes. That's not what I cut and pasted!
I'm trying to demonstrate. It shouldn't
be too hard to find.
Students: Ha, ha, ha. Ha, ha, ha. Sorry,
it's just the Yahoo thing. Sorry about
that.
Matt: I don't believe in psychics. But
they have a good random number generator.
Something didn't get copied right.
OK, I didn't copy that right. Then you go
to Microsoft Word and paste it. Go to re-
place. Type in "0 " and "0" in Find. Now
it will remove all of the spaces where
there's a 0. Now you have it at "1 " and
"1" in Find and it removes all of the
spaces that are behind 1 and you keep on
doing that. So you take - This was from
this morning. I was able to calculate a
prime with over 3,486 digits. I made the
same number of digits that will fit on a
page in Microsoft Word. So if you want to
find a near-random digit number. Type in,
this would be 2 lines, but it's such a
long equation that it's more. Type in
ABC2 2*$a+random_digit_number
and then you have an
a: from 0 to whatever_number_you_want
Save as worktodo.txt.
In Open PFGW, um, OK, and finally, let's
find a near-random digit prime and it
took only about a minute. OK. It act-
ually found more than 1. I did this
morning and it only found one. That is
how you find a near-random digit prime
with 3,486 digits. Alpertron is the
one I already showed you.
Alpertron.com.ar (for Argentina) /ECM.htm
That's the one that you need Java to use.
It works with Google Chrome and Mozilla
Firefox, but not always with Internet
Explorer for some reason. It's for
finding smaller primes with about 9,332
digits or less. Another program I use,
NewPGen, it uses the sieve of Erastos-
thenes. You've heard of Erastosthenes
from Carl Sagan's Cosmos. But imagine
that you take 400 million equations.
It eliminates millions of equations
instantly because it first eliminates
the ones that are multiples of 2, then
the ones that are multiples of 3, 5, 7,
11, and so on. This file is at 34
trillion. Awhile ago I typed in 2 for the
minimum k value and 400,000,001 for the
maximum k value. This was several days
ago. Sometimes I save the file after a
couple minutes. Sometimes I save the
file after a day. Where there's that k
it would be a number between 2 and 400
million. At the very beginning it
would eliminate millions in a second
several days ago. Now it's eliminating
1 k value about every 5 seconds. Since
I've gotten to this room about an hour
ago, it's eliminated more than 600
combinations. ...and then you run it
through Open PFGW. I have read that
more titanic or gigantic primes have
been discovered with Proth than any
other program, but I don't use Proth
much. PrimeForm is no longer updated
on the internet. Use Open PFGW from me,
or The Yahoo Group, or a
"download PFGW" internet search. GIMPS,
The Great Internet Mersenne Prime
Search and Prime95 are fast, but you
need to know how to use them. The Rep-
unit Project is an Italian web site but
they know perfect English.
They list the Repunits.
It's elek with a k. elektrosoft.it
then you click on Mathematica, and then
you click on Repunit. If you want to
start, then you email Maksym and Giovanni.
They are Mathematicians and I participate
in this project too. Now I use Windows XP
and 2 CPUs. It can test 2 repunits with
over 2 million digits in 2 or 3 days. It
tests them both at the same time because
you have 2 CPUs. Every digit in a repunit
is 1 and the number of digits is prime.
You don't have to learn very much more Math
or programming. It's very easy to do. Rep-
unit primes include 2 digits, 19 digits,
23 digits, 317, and 1,031. Let's have a
demo of that. If I just type in a random
number, it's not going to be very likely to
be prime. It's going to factor it. But I'll
type in a repunit number. 10^1031-1 and it
becomes all nines. Then you divide it by 9
and then you get a number that's all ones
digits. It's 1,031 digits. It's proving it
prime in base 2, 3, 5, 7, and so on. The
probable primes, because there's different
criteria for Chris Caldwell's primes web
site, The Top 5,000, than there is for The
Lifchitz brothers' web site. The Lifchitz
brothers' web site have 10,000 or more
digits and they use different equations.
But the probable primes have repunits, and
they include 49,081 digits, 86,453 digits,
109,297 digits, and 270,343 digits, and no
others below 2 million digits. I don't
actually use NVIDIA CUDA. If you play com-
puter games like Unreal Tournament 3, then
you have that graphics card. It works 6
times faster with NVIDIA CUDA. I don't have
time to show the videos but this presenta-
tion will be online. The videos are about
NVIDIA CUDA graphics cards. One of them's
just amazing. The computer recognizes the
faces of everybody in the hallway that's
walking by. With another NVIDIA CUDA com-
puter it actually recognizes the mosquitoes
and shoots them down with a laser. That's
done in poor areas where they don't believe
in pesticides to kill mosquitoes.
Students: Ha, ha, ha. Ha, ha, ha.
Matt: That's with NVIDIA CUDA. I have dis-
covered since February, 4 more probable
primes. I've sieved in NewPGen,
210^n+199, after a couples days I ran it
through Open PFGW. That's based on a Dis-
cover magazine formula. Also, another theme
that I was thinking about was a 10k equation
because there was 10K web design project. It
was a contest to make the best web site that
you can in 10k, or 10 kilobytes, that would
be 10,240 bytes. So, I found the largest
prime number with 10,240 digits. Another
theme that I was thinking about was with the
Windows calculator. OK, this is different
than what I have at home, but - OK, If you
take 2^144269 - See, OK, they've changed it.
I guess that the one with Windows XP is
different than the one with Windows 7.
Student: They can probably use Wolfram?
Amitavi: Wolfram Alpha?
Matt: Oh.
Anyways, so with the Windows XP ver-
sion if you try 2^144269, you get a 43,430-
digit number. If you try one more in the
exponent, 144270, then you get an error. I
set up NewPGen for days the 43,430-digit
equations. Then I switched to Open PFGW.
It discovered about every 2 days, a prime
number that is probable with 43,430
digits. So I'll show you how to submit
these. If you search "Probable Primes"
Lifchitz L i f c h i t z and then hit enter,
you get the web site. Then you click on
"Submit your PRPs." You enter the verifi-
cation code or captcha that's to show that
you're not a machine. Choose your name if
you've already submitted. These are the 4
equations that I've just discovered since
February that I've just talked to you about.
You can use your email address for confirm-
ation. So, I just submit it. Well, I don't
know what the problem is. But it will appear
on the web page in a couple days. Just sub-
mit a prime, a probable prime, and it will
show up in a couple days. So there are 5
that are discovered with a million or more
digits. I'm almost finished. I thought that
I'd get the question, "What's the largest
candidate prime number that you've tested?"
I have tested numbers with over 2 million
digits and the largest probable prime I've
discovered has 43,430. However, Repunit
Team member Danilo from Germany has tried
to test a mole number of digits of
2^(2 septillionth)-1, that's Avogadro's
number of digits. It has no factors of less
than 2^120 in Factor5, that is a 64-bit pro-
gram. I don't have 64-bit Windows. I've
shown you Open PFGW, but that's the Win-
dows version. I don't know how it's differ-
ent in Linux or Macintosh. Does
anybody have any questions?
Kilian: A lot of these formulas are 2 raised
to some power and you do you other stuff.
Why is it 2 as a base instead
of 3 or some other number?
Matt: Because with 2 you get to find and
eliminate factors faster. It runs, I think
it's called Lucas-Lehmer tests, which I don't
fully understand. You're talking about the
Mersenne primes, there are
almost 50 of them. Questions?
Student: In terms of speed, what if you used
3 as an extention to some power? How would
that effect the speed of eliminations? Well,
You'd have to have used NewPGen to sieve out
the numbers. First of all, if you have 3 to
a power and then you subtract 1, it's going
to be even. So you'd have to have a coeffi-
cient in front of it. You have an option to
use a base of 3 and use whatever exponent
you want but then if it's an odd base, and
then you add 1 or you subtract 1. Then the
coefficient has to be even. I've gotten the
question, "Are some Mersenne numbers more
difficult to test than others?" If a Mer-
senne prime or a prime candidate has X
times as many digits, then it would take
X squared times as long to prove. With 3
times as many digits, it would take 9
times as long to prove. With 4 it would be
16 and so on. How many primes and perfect
numbers with more than 1 million digits are
there? 76 for now. There are 11 Mersenne,
51 prime non-Mersenne, 5 probable primes,
and 9 perfect numbers with 1 million or
more digits. Perfect numbers are not prime.
Perfect numbers are like 6 or 28. It adds
up to its factors. The factors of 6 are 1,
2 and 3, and that adds up to 6.
Amitavi: Not itself though.
Matt: What!?
Amitavi: Not itself.
Professor Wagstaff: Right. That's right.
Matt: How many primes are in The Chris
Caldwell's and The Lifchitz brothers' data-
bases? Chris Caldwell has over 106,000 and
the Lifchitz database has over 94,000.
...and as far as books, I recommend The
Little Book of Big Primes from about 1991,
The Little Book of Bigger Primes from about
2004, and Prime Curios from interlibrary
loan. I haven't been able to get Prime
Curios yet but I think that you can get
both The Little Book of Big Primes and The
Little Book of Bigger Primes from Purdue.
...and write down my email. It's stathmk
at yahoo.com if you're interested in find-
ing prime numbers with 10,000 or more
digits. Contact me first, and if I don't
know, then I'll contact PrimeForm, PFGW,
PrimeGrid, and The Mersenne Forum. Briefly
email me if you find a Top 5,000 prime or
a probable prime with 10,000 or more
digits. This will be on Youtube in April
with a link to The Powerpoint.
Any more questions?
Kilian: You have several prime formulas,
like the Euler formula x^2+x+41. Is it
known whether that formula can produce
infinitely many primes?
Matt: I did not think about that.
Professor Wagstaff: No, I
don't think that is known.
Matt: Somebody proved that that are in-
finitely many arithmetic progressions'
primes. I thought I heard that there are
infinitely many twin primes.
Amitavi: No, that's not true either.
That's an open problem. Twin primes.
Matt: ...and if the gigantic primes are
too much for you, then try looking for
BiTwin primes. BiTwin primes sets are
like 30 and 60 because you double it you
get 60. 29, 31, 59, and 61 are prime. I
recommend The Repunit Project
to participate in.
Amitavi: BiTwin?
Matt: So, they have apparently not dis-
covered more than 50 BiTwin sets. That
would be where you double it, and the one
that's below it, and the one right above
it are prime. I have a web site. Matt
Stath .com. That's Matt s t a t h .com,
and then you click on Top Primes. If
anybody figures out a theme for primes,
or a group, or they have their idea for
a Top 50 list, then it's mattstath.com.
Is there anything else?
Amitavi: What is your favorite prime?
Matt: What is my favorite prime? It
might be Jenny's number because of the
song, "867-5309."
Amitavi: What's that number?
Matt: It's Tommmy Tutone. He sings
that song. Where he meets a girl and -
Amitavi: Phone number?
Matt: - and her phone number is
867-5309. That's a prime number.
Amitavi: 867-5309!
Matt: 867-5309. ...and I
think we're done.
Kilian: Any more questions?
Classroom: Clapping.
Matt: Thank you.