Thursday, 12 April 2018
Monday, 9 April 2018
Largest known prime number
Largest known prime number
discovered with over 23 million digits
Discovery made on computer
belonging to electrical engineer who searched for the elusive number for
14 years
A collaborative computational
effort has uncovered the longest known prime number.
At over 23 million digits long,
the new number has been given the name M77232917 for short.
Prime
numbers are divisible only by themselves and one, and the search
for ever-larger primes has long occupied maths enthusiasts.
However, the search requires
complicated computer software and collaboration as the numbers get
increasingly hard to find.
M77232917 was discovered on a computer
belonging to Jonathan Pace, an electrical engineer from Tennessee who has been
searching for big primes for 14 years.
Mr Pace discovered the new
number as part of the Great Internet Mersenne Prime Search (GIMPS), a project
started in 1996 to hunt for these massive numbers.
Mersenne primes – named after
the 17th century French monk Marin Mersenne who studied them – are calculated
by multiplying together many twos and then subtracting one.
Six days of non-stop computing
in which 77,232,917 twos were multiplied together resulted in the latest
discovery.
The number is the 50th Mersenne
prime to be discovered, and the 16th to be discovered by the GIMPS project.
It is nearly one million digits
longer than the previous record holder, which was identified as part of the
same project at the beginning of 2016.
Mersenne primes are a
particular focus for prime aficionados because there is a relatively
straightforward way to check whether a number is one or not.
Nevertheless, the new prime has
to be verified using four different computer programs on four different
computers.
The process also relies on
thousands of volunteers sifting through millions of non-prime candidates before
the lucky individual chances upon their target.
“I’m
very surprised it was found this quickly; we expected it to take longer,”
Professor Chris Caldwell, a mathematician at the University of Tennessee at
Martin told The
Guardian.
Professor Caldwell runs an
authoritative website on the largest prime numbers, with a focus on the history
of Mersenne primes.
He emphasised the pure
excitement that searching for prime numbers brings, describing the latest
discovery as “a museum piece as opposed to something that industry would use”.
Besides the thrill of
discovery, Mr Pace will receive a $3,000 (£2,211) GIMPS research discovery
award.
GIMPS uses the power of
thousands of ordinary computers to search for elusive primes, and the team
behind it state that anybody with a reasonably powerful PC can download the
necessary software and become a “big prime hunter”.
The next Mersenne prime
discovery could be smaller or larger than the existing record holder, but the
big target for the GIMPS team is to find a 100 million digit prime number.
The person who discovers such a
number will be awarded $150,000 by the Electronic Frontier Foundation for
their efforts.
Friday, 23 February 2018
20 Things You Didn't Know About... Math
1 The median score for college-bound seniors on the math section of the SAT in 2011 is about 510 out of 800. So right there is proof that there are lots of unsolved math problems.
2 The great 19th-century mathematician Carl Friedrich Gauss called his field “the queen of sciences.”
3 If math is a queen, she’s the White Queen from Alice in Wonderland, who bragged that she believed “as many as six impossible things before breakfast.” (No surprise that Lewis Carroll also wrote about plane algebraic geometry.)
4 For example, the Navier-Stokes equations are used all the time to approximate turbulent fluid flows around aircraft and in the bloodstream, but the math behind them still isn’t understood.
5 And the oddest bits of math often turn out to be useful. Quaternions, which can describe the rotation of 3-D objects, were discovered in 1843. They were considered beautiful but useless until 1985, when computer scientists applied them to rendering digital animation.
6 Some math problems are designed to be confounding, like British philosopher Bertrand Russell’s paradoxical “set of all sets that are not members of themselves.” If Russell’s set is not a member of itself, then by definition it is a member of itself.
7 Russell was using a mathematical argument to test the outer limits of logic (and sanity).
8 Kurt Gödel, the renowned Austrian logician, made matters worse in 1931 with his first incompleteness theorem, which said that any sufficiently powerful math system must contain statements that are true but unprovable. Gödel starved himself to death in 1978.
9 Yet problem solvers soldier on. They struggled for 358 years with Fermat’s last theorem, a notoriously unfinished note that 17th-century mathematician and politician Pierre de Fermat scrawled into the margin of a book.
10 You know how 32 + 42 = 52? Fermat claimed that there are no numbers that fit the pattern (an + bn = cn) when they are raised to a power higher than 2.
11 Finally, in 1995, English mathematician Andrew Wiles proved Fermat was right, but to do it he had to use math Fermat never knew existed. The introduction to Wiles’s 109-page proof also cites dozens of colleagues, living and dead, on whose shoulders he stood.
12 At a conference in Paris in 1900, German mathematician David Hilbert determined to clear up some lingering math mysteries by setting out 23 key problems. By 2000 mathematicians had solved all of the well-formed Hilbert problems save one–a hypothesis posed in 1859 by Bernhard Riemann.
13 The Riemann hypothesis is now regarded as the most significant unsolved problem in mathematics. It claims there is a hidden pattern to the distribution of prime numbers—numbers that can’t be factored, such as 5, 7, 41, and, oh, 1,000,033.
14 The hypothesis has been shown experimentally to hold for the first 100 billion cases, which would be proof enough for an accountant or even a physicist. But not for a mathematician.
15 In 2000 the Clay Mathematics Institute announced $1 million prizes for solutions to seven vexing “Millennium Prize Problems.” Ten years later the institute made its first award to Russian Grigori Perelman for solving the Poincaré conjecture, a problem dating back to 1904.
16 Proving that mathematicians don’t grasp seven-digit numbers, Perelman turned down the million bucks because he felt another mathematician was equally deserving. He currently lives in seclusion in Russia.
17 In his teens, Evariste Galois invented an entirely new branch of math, called group theory, to prove that “the quintic”—an equation with a term of x5—was not solvable by any formula.
18 Galois died in Paris in 1832 at age 20, shot in a duel over a woman. Anticipating his loss, he spent his last night frantically making corrections and additions to his math papers.
19 Graduate student George Dantzig arrived late to statistics class at Berkeley one day in 1939 and copied two problems off the blackboard. He handed in the answers a few days later, apologizing that they were harder than usual.
20 The “homework” was actually two well-known unproven theorems. Dantzig’s story became famous and inspired a scene from Good Will Hunting.
Friday, 5 January 2018
Tuesday, 2 January 2018
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Interesting Math Problem
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|
Combinations - 10-digit numbers of 1s,2s,3s
a) How many different
10-digit numbers can be formed from the digits 1, 2, 3 where digit 3 in each
number is found exactly three times ?
b) How many of those numbers can be divided by 6 ?
b) How many of those numbers can be divided by 6 ?
a) How many different
10-digit numbers can be formed from the digits 1, 2, 3 where digit 3 in each
number is found exactly three times ? Choose the three spots for the 3's in C(10,3) ways. 10 × 9 × 8 /
3!
For the remaining 7 spots, they can be either a 1 or a 2, so there are 2^7 possibilities.
In total we have (10 × 9 × 8 / 3!) × 2^7 = 15360
b) How many of them are divisible by 6 ?
In order to be divisible by 6, the number must be even and divisible by 3.
If the only available digits are 1,2,3, then to be even, the number must
end in 2.
To be divisible by 3, the sum of the digits must also be divisible by 3.
So we start with three 3's and a 2.
The sum of the remaining six digits must have a remainder of 1 (mod 3).
For the sum, 1 and 4 are < 6 × 1 and 13 > 6 × 2, so the only
possibilities are 7 and 10.
The remaining six digits (in any order) can be:
2 2 2 2 1 1 sum of 10
2 1 1 1 1 1 sum of 7
So we combine each of these with the three 3's,
and we get
9! /(3! 4! 2!) + 9! / (3! 1! 5!) = 1260 + 504 = 1764
(The 10th digit is always a 2.)
Another way of looking at this is:
first assign the threes to C(9, 3) places,
then assign the twos to C(6, 4) or C(6, 1) places.
Then we get
9 × 8 × 7 / 3! × (6 × 5 × 4 × 3 / 4! + 6 / 1!) = 84 × 21 = 1764
For the remaining 7 spots, they can be either a 1 or a 2, so there are 2^7 possibilities.
In total we have (10 × 9 × 8 / 3!) × 2^7 = 15360
b) How many of them are divisible by 6 ?
In order to be divisible by 6, the number must be even and divisible by 3.
If the only available digits are 1,2,3, then to be even, the number must
end in 2.
To be divisible by 3, the sum of the digits must also be divisible by 3.
So we start with three 3's and a 2.
The sum of the remaining six digits must have a remainder of 1 (mod 3).
For the sum, 1 and 4 are < 6 × 1 and 13 > 6 × 2, so the only
possibilities are 7 and 10.
The remaining six digits (in any order) can be:
2 2 2 2 1 1 sum of 10
2 1 1 1 1 1 sum of 7
So we combine each of these with the three 3's,
and we get
9! /(3! 4! 2!) + 9! / (3! 1! 5!) = 1260 + 504 = 1764
(The 10th digit is always a 2.)
Another way of looking at this is:
first assign the threes to C(9, 3) places,
then assign the twos to C(6, 4) or C(6, 1) places.
Then we get
9 × 8 × 7 / 3! × (6 × 5 × 4 × 3 / 4! + 6 / 1!) = 84 × 21 = 1764
Thursday, 7 December 2017
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