Slashdot Mirror
New Mersenne Prime Discovered, Largest Known Prime Number: 2^74,207,281 - 1 (mersenne.org)
Dave Knott writes: The Great Internet Mersenne Prime Search (GIMPS) has discovered a new largest known prime number, 2^74,207,281-1, having 22,338,618 digits. The same GIMPS software recently uncovered a flaw in Intel's latest Skylake CPUs, and its global network of CPUs peaking at 450 trillion calculations per second remains the longest continuously-running "grassroots supercomputing" project in Internet history. The prime is almost 5 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 49th known Mersenne prime ever discovered, each increasingly difficult to find.
Here you go: S. Wagstaff, "Divisors of Mersenne numbers," Math. Comp., 40:161 (January 1983) 385--397. MR 84j:10052
It's true that we don't know for sure, but it's not true that we have no fucking idea.
sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});
this is likely true as the number 1 is not a prime number. https://primes.utm.edu/notes/f...
No, absolutely not.
Out of 74,207,281(-ish) tested values for n, this is only the 49th prime found. If you tested the first 74,207,281 odd numbers you would have found more that 5 million primes.
09F91102 no, 455FE104 nope, F190A1E8 uh-uh, 7A5F8A09 that's not it, C87294CE no. Ah! 452F6E403CDF10714E41DFAA257D313F.
It's not so much that Mersenne numbers are much more likely to be prime than other odd numbers of their size. It's that there is a special-purpose primality test just for Mersenne numbers that is tons more efficient than verifying other primes of similar size.
Isn't knowing the definition of a prime number enough?
Enough - for what? The definition of prime numbers is deceptively simple, but we still don't know a general way to construct all prime numbers - we don't even know if there is one. The same can be said for many other classes of numbers, I suppose, but prime numbers have turned out to be useful for our understanding of numbers and other things.
Compare with vector spces: a vector space is, to put it simply, a space with 'dimension': every point in a vector space can be represented as a tuple of numbers: a = (a1, a2, a3, ...., aN). The first thing you want to find in a vector space is the basis: a set of N vectors that point out the independent coordinate axes of the space think of R^2 or R^3, the 2- and 3-dimensional spaces we are familiar with. Every natural number has a slightly similar property: it can be written as a product of prime numbers - the prime factors. This can useful when you calculate things - if you know that 30030 = 2*3*5*7*11*13 and 136367 = 7*7*11*11*23, then it is easy to see that 136367/30030 = 2*3*5*13/7*11*23; sometimes it is easy to find prime factors, at least if you know what the prime numbers are.
Also, in the theory for finite groups, if p is a prime number, then any group with p elements is cyclic and any group with p^2 is Abelian (wikipedia is your friend, if you want to know more); cryptic, I know, but it has profound consequences.
Math is also fascinating because of how it can often work around impossibility proofs.
E.g., what class of polynomials is solvable depends on what elementary functions are allowed. With Jacobi theta functions, you can exactly solve quintics.
http://mathoverflow.net/questi...
For another example, with cosine and acos, you can exactly solve cubic polynomials, w/o using cube roots. Better, if the solutions are real, then the solution does not require imaginary numbers, unlike if you solve with cube roots.