Matt Stath GIGANTIC PROBABLE PRIMES PURDUE MATH CLUB

Largest Known Prime � From Warner Bros on Youtube. � From The. Singing. Nerd on Youtube. � From Student CNN News on Youtube. � 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 a box of printer paper!

5/6 Of This Box!

2 nd Largest Known Prime � From The Associated Press on Youtube. � 2^43, 112, 609 -1 � 12, 978, 189 digits divided by 3, 486 digits per page is almost 3, 723 pages! Printed on both sides of the paper, it would be about 1, 861 ½ pages or 5/8 of a box of printer paper! � These use GIMPS.

100 Million Digits!? � In case you are wondering from the previous video, the first 100 million digit base 2 number would be 2^332, 192, 807 -1. � But it’s not very likely to be prime. � 100 million digits divided by 3, 486 digits per page is almost 28, 687 pages! Printed double sided it would be 14, 344 pieces of paper or 4 4/5 boxes of paper!

Origins of Primes � The Ishango Bones of baboons discovered in 1960 in Belgian Congo from about 20, 000 BC. …. Or the Lebombo Bones of Swaziland in 35, 000 BC?

Origins of Primes 2 � Let’s take a closer look.

Origins of Primes 3 � The Sumerian Cuneiform in ancient Iraq had a list of primes according to Vern Foley, one of my History professors. They were base 60.

Origins of “Titanics” � Archimedes The Great’s “Myriad of Myriads” is 100, 000^100, 000 or 1 followed by 80 million billion zeros. It isn’t prime. � He believed in The Sun god or Sun titan Hyperion 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!?

Origins of “Titanics” Part 2 � Praise Hyperion! � However, it may have been propaganda against Archimedes because he was a different type of Greek than the Romans cared for.

Origins of “Titanics” Part 3 “The Mathematics Teacher” Newsletter in 1971 theorized a 1, 560 -digit 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. �

Origins of “Titanics” 4 There are believed to be 10^2, 400, 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) in humans because of the G, C, A, & T base pairs. �

How I Became Interested �I was in the 99 percentile in Math, 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 3 rd grade & I thought that’s the stupidest thing that they said on the show all year!

How I Became Interested 2 Math. Counts in 7 th & 8 th grades. � In 2003 -04 reading in The Exponent, I saw 2 computer articles that caught my eye: � A Leukemia cure crowd sourcing screensaver (probably a precursor to Fold It Beta) � Purdue students tried an equation to find a prime with over 40, 000 digits and it didn’t work. Were they Chen & Wang? ? � Also, a 2005 Discover Magazine Article. �

Prime Web Pages The Top 5, 000 at http: //primes. utm. edu/ The prime in 5, 000 th place had about 70, 000 digits that year that I saw the Discover article. � The Top 20 Lists at http: //primes. utm. edu/top 20/index. php � GL Honaker Jr. ’s Curios at http: //primes. utm. edu/curios � Also, there’s Lifchitz’ Probable Primes at http: //www. primenumbers. net/prptop. php (These have 10, 000 or more digits. ) � �

Why Are Primes Important? Public key cryptography: P 1*P 2=C P 1 = Private key P 2 = Private key C = Public key � Education, hypotheses, Autistic savants like Daniel Tammet, Ulam's Spiral, theorems, formulas, SETI, Mersenne Primes, bee & rabbit family trees, contests, prizes, & etc! �

Why Are Primes Important? Part 2 � Mersenne family trees: 16 seconddegree great grandparents + 8 great grandparents + 4 grandparents + 2 parents + yourself = 31. � Fibonacci family trees in bees and rabbits � Rabbits have a gestation of a month, give birth, and are 1, 1, 2, 3, 5, 8, or 13 pairs in the previous month.

Why Are Primes Important? Part 3: Fibonacci Sequence

Prime Generating Formulas � I’ll put variable n or x in red in these slides. � From Wolfram: x^2+x+41 is prime for x=0 to 39 (Euler Formula) � From Discover: 210*n+199 for n=0 to 9 � Discover: 4, 609, 098, 694, 200*n+ 11, 410, 337, 850, 553 for n=0 to 21 � Discover: 18, 549, 279, 769, 020*n+ 376, 859, 931, 192, 959 for n=0 to 21

x^2+x+41 Part 1 � The first prime that I care to remember discovering was 999 digits and was (10^499)^2+(10^499)+41 in Alpertron. I was so excited that I submitted it to The Prime Curios. � As time went on, I discovered more with the formula in the Prime. Form program: � (10^10889)^2+(10^10889)+41 (21179 Digits)

x^2+x+41 Part 2 � (20731#/41)^2+(20731#/41)+41 (17820 Digits) Note: # is primorial. 20731 primorial is (2*3*5*7*11*13*17*19*23*29*…*20731) � (2^19780)^2+(2^19780)+41 (11909 Digits) � (10^5059)^2+(10^5059)+41 (10119 Digits)

x^2 -x+41 Part 1 � (2^33161)^2 -(2^33161)+41 (19965 Digits) � (2^32314)^2 -(2^32314)+41 (19455 Digits) � (10^8682)^2 -(10^8682)+41 (17364 Digits) � (10^8506)^2 -(10^8506)+41 (17012 Digits)

x^2 -x+41 Part 2 � (10^8046)^2 -(10^8046)+41 (16092 Digits) � (2^23462)^2 -(2^23462)+41 (14126 Digits) � (10^6007)^2 -(10^6007)+41 (12014 Digits)

210*n+199 23363#/(199*11)+199+210*10343 (10081 Digits) � 23363#/(199*11)+199+210*5283 (10081 Digits) � 23363#/(199*11)+199+210*3302 (10081 Digits) � 23363#/(199*11)+199+210*367 (10081 Digits) � 23363#/(199*11)+199+210 (10081 Digits) �

Discover Magazine Equations � 1854927976902*10^21813+376859931 192959 (21826 Digits) � 46090986942*10^19644+11410337850 553 (19655 Digits) � 46090986942*10^11866+11410337850 553 (11877 Digits)

Wikipedia & Prime. Grid Equation � 23681770*23#*n+43142746595714191, for n = 0 to 25 � 24023#*(7#/3)+43142746595714191 (10384 Digits)

Generalized Fermat numbers usually are k*(even base)^(2^x)+1. � Formula: k*(even base)^(2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, or 524288)+1. � Somebody discovered one last year with an exponent of 524288 and almost 3 million digits. � I’m trying top find a Top 5000 Generalized Fermat in New. PGen with over 410, 000 digits! �

Generalized Fermat with a “+2” Part 1 � 61461^4096+2 (19615 Digits) � 27189^4096+2 (18164 Digits) � 2085^4096+2 (13596 Digits) � 447147^2048+2 (11573 Digits) � 335073^2048+2 (11316 Digits) � 329961^2048+2 (11302 Digits) � 285603^2048+2 (11174 Digits) � 239223^2048+2 (11016 Digits)

Generalized Fermat with a “+2” Part 2 � 158667^2048+2 (10651 Digits) � 92169^2048+2 (10168 Digits) � 89667^2048+2 (10143 Digits) � 5816368677^1024+2 (10000 Digits) � 33829600955336781639^512+2 (10000 Digits) � 11444419007973230259545322536495 98074879^256+2 (10000 Digits)

Generalized Fermat with a “+2” Part 3 � 13097472643005897581100324196733 47620959731068338635399869049806 286716959555931^128+2 (10000 Digits) � For now, Henri and Renaud Lifchitz will allow an equation with up to 100 characters on their web site.

Probable Prime Palindromes �A palindrome is a number or phrase like, “Ana, nab a banana” or “A man, a plan, a canal, panama” that reads the same forwards and backwards. � (((10^111)/9)*10^20+111*10^14)*(10^(1180*20)1)/((10^20 -1)/1)+((10^11 -1)/9) is a 23, 611 digit probable prime palindrome I’ve discovered! (Demonstration)

Random and Picture Primes � I’ve made a Dopefish Probable Prime.

Random and Picture Primes 2 � On Matt Stath. com, you can email me your near-random primes, picture primes, and primes of over 100 characters. � That’s so you can link to them on one central page from Henri and Renaud's Probable Primes when you submit. � That’s because they’ll only display equations of up to 100 characters for now.

Programs I Use � � � Alpertron factors “smaller” numbers and proves them in Google Chrome. New. PGen sieves 400 million equations at a time. (Demo of Sieve of Erastosthenes) I had read that more titanic or gigantic primes have been discovered with Proth than any other program, but I don’t use it much. Prime. Form is no longer updated on the internet. Use Open PFGW from me, or The Yahoo Group, or a "download PFGW" internet search. GIMPS and/or Prime 95 are fast, but you need to know how to use them. (Demonstration)

Repunit Project � Email Giovanni and Maksym to join The Repunit Project. � With Windows XP and 2 CPUs, you can test 2 repunits with over 2 million digits in 2 or 3 days. � 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.

Repunit Project Part 2 � Repunit primes include (10^2 -1)/9, (10^19 -1)/9, (10^23 -1)/9, (10^317 -1)/9, and (10^1031 -1)/9. (Demo) � The probable primes include (10^490811)/9, (10^86453 -1)/9, (10^109297 -1)/9, (10^270343 -1)/9, … � …and no others below 2 million digits. � Proving works 6 times faster with an NVIDIA CUDA graphics card. Video 1 & Video 2

Tonight’s Demonstration of PRP Submissions 210^8802+199 (20441 Digits) (Based on a Discover formula) � 10^(10*1024)-2*66851 -1 (10240 Digits) (a 10 K equation) � 2^144269+2*76811 -1 (43430 Digits) (a Windows XP calculator prime) � 2^144269+2*105936 -1 (43430 Digits) (a Windows XP calculator prime) � Time for a Q & A session. �

Q & A Session Part 1 Q: What’s the largest candidate number you’ve tested? � A: I have tested numbers with over 2 million digits and the largest probable prime I’ve discovered has 43, 430 digits. � However, Repunit Team member Danilo in Germany has tried to test a mole number of digits of 2^20007972915506583447722271. It has no factors of less than 2^120 in Factor 5. �

Q & A Session Part 2 � Q: Are some Mersenne numbers more difficult to test than others? � A: If a Mersenne prime has X times as many digits, then it takes X^2 times as long to prove. With 3 times as many digits, it would take 9 times as long to prove. With 4 times as many digits, it would take 16 times as long to prove.

Q & A Session Part 3 � Q: How many primes and perfect numbers with more than 1 million digits are there? � A: 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.

Q & A Session Part 4 � Q: How many primes are in Chris Caldwell’s and The Lifchitz Brothers’ databases? � A: Chris Caldwell’s database has over 106, 000 primes. The Lifchitz database has over 94, 000 probable primes.

Q & A Session Part 5 � Q: What books do you recommend? � A: I recommend The Little Book of Big Primes (about 1991), The Little Book of Bigger Primes (about 2004), and Prime Curios from interlibrary loan.

Formulas, Questions, or Help? � Contact Matt Stath first at stathmk@yahoo. com. � Then contact The Purdue Math Club, Prime. Form & PFGW user group, Prime. Grid, and The Mersenne Forum. � Briefly email me if you find a Top 5, 000 Prime or a Probable Prime. � This will be on Youtube in April with a link to the Powerpoint.