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The largest prime number currently known is
2
57,885,161
1
(It has 17,425,170 decimal digits)
The largest prime number currently known is
2
57,885,161
1
(It has 17,425,170 decimal digits)
It was discovered on January 25, 2013 as part of
the Great Mersenne Prime Internet Search project.
This project now is running on over 350,000 processors
and performing more that 150,000,000,000,000
computations per second.
The largest prime number currently known is
2
57,885,161
1
(It has 17,425,170 decimal digits)
It was discovered on January 25, 2013 as part of
the Great Mersenne Prime Internet Search project.
This project now is running on over 350,000 processor
and performing more that 150,000,000,000,000
computations per second.
A prize of $150,000 is being offered by the
Electronic Frontier Foundation for the first
prime discovered which has over 100,000,000
decimal digits.
Recent progress (Helfgott - 2013)
Every odd number greater than 5
is the sum of three primes.
Recent progress (Helfgott - 2013)
Every odd number greater than 5
is the sum of three primes.
(The previous bound was 2 x 101346 ).
The largest known twin primes currently known
are
3756801695685 2
666669
They each
have
200, 700
decimal
digits !
1
The largest known twin primes currently known
are
3756801695685 2
666669
They each
have
200, 700
Remarkable new theorem
1
decimal
digits !
(Zhang
Yitang
張益唐(2013))
There is a number N < 70,000,000 such that
there are infinitely many pairs of prime numbers of
the form p and p + N.
The largest known twin primes currently known
are
3756801695685 2
666669
They each
have
200, 700
Remarkable new theorem
1
decimal
digits !
(Zhang
Yitang
張益唐(2013))
There is a number N < 70,000,000 such that
there are infinitely many pairs of prime numbers of
the form p and p + N.
The current record is
N  270
(as of April 2014)
( $1,000,000 )
until recently.
This is definitely not best possible!
Some combinatorics....
_
NP-complete
 (G)=3?  (G)=k?
Bin Packing
Traveling Salesman
NP
+ Thousands of other
important algorithmic
problems .....
Graphs G  H ?
Is N prime ?
Convex Hull
Sorting
Linear Programming
P
Landscape of NP
3.
No one knows!
=
=
This is problem No. 1 on a recent list
of $1,000,000 problems
$1,000,000
=
This is problem No. 1 on a recent list
of $1,000,000 problems
$1,000,000
Most experts conjecture that the answer is No.
SI
X
?
A double torus unknotted!
A double torus unknotted!
A double torus unknotted!
$ 250
It can be shown that the greedy
decomposition for the fraction
______
5
1444613
terminates in 37 steps.
It can be shown that the greedy
decomposition for the fraction
______
5
1444613
terminates in 37 steps.
It has a largest denominator with
384,122,451,172 decimal digits!
It can be shown that the greedy
decomposition for the fraction
______
5
1444613
terminates in 37 steps.
It has a largest denominator with
384,122,451,172 decimal digits!
What is this trying to tell us ?
$25 Does the greedy algorithm always terminate ?
$25 Does the greedy algorithm always terminate ?
$25 Does the greedy algorithm always terminate ?
.
22 Æ 33 - 1 = 2³32 + 2³3 + 2 Æ 2³ 42 + 2³ 4 + 1 Æ 2³52 + 2³5 Æ 2³62 + 2³6 - 1 = 2³62 + 1 ³6 + 5 Æ ......
$ 1,000,000
$ 1,000,000
$ 1,000,000
not yet
in
,.! not numbers.
A Cynic Richard Hamming
(from a speech of David Hilbert on Sept. 8, 1930 )
“Then I say to myself, ‘What’s the use?
There isn’t any Nobel Prize for math’.”
But there is the Abel Prize !
The 2014 Abel Prize
Winner Yakov G. Sinai