12 votes

Prime Spirals, by Numberphile, with Dr. James Grime


http://youtu.be/iFuR97YcSLM

Ulam Spiral

The Ulam spiral, or prime spiral (in other languages also called the Ulam Cloth) is a simple method of visualizing the prime numbers that reveals the apparent tendency of certain quadratic polynomials to generate unusually large numbers of primes. It was discovered by the mathematician Stanislaw Ulam in 1963, while he was doodling during the presentation of a "long and very boring paper" at a scientific meeting [...]

Read on :

http://en.wikipedia.org/wiki/Ulam_spiral




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Cyril's picture

Terence Tao's Structure and Randomness in the Prime Numbers

Another classic, already, to know about:

Terence Tao's Structure and Randomness in the Prime Numbers

( His home page : http://www.math.ucla.edu/~tao/ )


http://youtu.be/hYxBH1YY9z4

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

jrd3820's picture

Bump

for spirals :)

Cyril's picture

.

.

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Cyril's picture

Confession

Primes investigation is one of my guilty pleasures as an amateur, right next to sipping some liquor in the evening.

;p

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Looking for a rational pi approximation better than (2*11)/7

5*71 / 113 =3.1415929203539823008849557522124
pi ====== 3.1415926535897932384626433832795...

err = 2.6676418906242231236893288649633e-7

There is a problem with the program I wrote, not reporting error correctly. Hmmm.

I enjoy a Brandy in the evening.

Free includes debt-free!

Cyril's picture

Check this out :

Check this out :

Say you're doodling around the semiprimes factorization problem and you aim for something much simpler (and less powerful) than the GNFS but still more efficient than trial division.

Let's get our hands on, say :

s = 136,129,581,883

to factor.

Of course, trial division tells us we can stop at int(sqrt(136,129,581,883)) = 368,957, and since all primes are known to be of either forms 6n + 1 or 6n - 1, that still leaves us with 368,957 / 6 = 61,492 attempts.

What if I told with you with some smart (and lucky ;) guess you can reduce it to less than 400 iterations only?

Here it is:

Let s = u . v, with u < v.

Certainly, we can always write:

(1) k . u < v < (k + 1) . u

For some natural k, 1 <= k <= N (that's our guess)

We can assume N rather small (< 100 or 10 or less), when the numbers of digits of u and v differ by one or two only (e.g., for a factor 1 to 100 between u and v).

Then, (1) times u:

k . u^2 < u . v < (k + 1) . u^2

That is:

umin < u < umax

with umin = int(sqrt(s / (k + 1))) and umax = int(sqrt(s / k))

Application:

create N copies (say, N = 12 ... I said we can be lucky ;) of the same program P(k), parameterized by k, 1 <= k <= N

which will write u = 6 . p +/- 1 growing from umin towards umax, e.g., for P(k = 12):

umin = int(sqrt(136,129,581,883 / 13)) = 102,331

umax = int(sqrt(136,129,581,883 / 12)) = 106,507

and will be testing s / u

In our case, it turns out that:

u = 6 * 17,455 - 1 = 104,729 (and v = s / u = 1,299,827)

where u (or p, equivalently) will be stumbled upon by the program P(k = 12) after:

(u - umin) / 6 = (104,729 - 102,331) / 6 = 399 iterations.

:)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Question

"since all primes are known to be of either forms 6n + 1 or 6n - 1"

Can you expand on this a little bit? I've never heard this before and I'm intrigued

Cyril's picture

Two lists of small primes I use regularly:

Two lists of small primes I use regularly:

http://primes.utm.edu/lists/small/100000.txt

http://prime-numbers.org/

For testing primality, Wolfram MathWorld comes in handy :

http://www.wolframalpha.com/input/?i=Is+123571113171923+prim...

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Cyril's picture

This is true for all primes greater than 3, starting with 5.

This is true for all primes greater than 3, starting with 5 (my inaccuracy by omission). Because all integers >= 5 can be expressed as (6k + i) for some integer k >= 1 and for i = −1, 0, 1, 2, 3, or 4.

Where: 2 divides (6k + 0), (6k + 2), (6k + 4); and 3 divides (6k + 3). Hence, only i = -1 or 1 are left.

There are many basic properties of prime numbers to be aware of, for anyone interested in them.

This is a great resource to start with:

http://primes.utm.edu/

(among others, often referenced from Wikipedia)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Cool

Seems so obvious in retrospect. Thanks!

Are you talking about the Golden ratio, or the golden mean?

and if so, why did you not present it as such?

Cyril's picture

Golden? Neither, as I have more of a thing for the silver

Golden? Neither, as I have more of a thing for the silver ratio and Pell numbers, lately:

http://en.wikipedia.org/wiki/Silver_ratio

;)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius

Seriously Dude? are you that numb?

both the Golden ratio AND the Golden mean have actual applications in the real world.
care to make the same statement for your conjecture?

and no sir, I am NOT a mathematician.. or a mathamagician.

how is your beloved "silver ratio" relevant? in any way, shape or form?

you consider me a stupid HVAC/R Tech. and you don't even know what the profession entails.

let me know how that works out for you.

what I AM aware of, is that there are MANY things that we (as humans)
know how to use and make them do work for us.

that we do not fundamentally understand.

peace.

The ancients knew Wau

http://www.youtube.com/watch?feature=player_embedded&v=GFLko...

Thanks Cyril, been listening to Math you tubes for two days now.

C'est fantastique, mon ami!

Free includes debt-free!

Cyril's picture

LOL 'Welcome :)

LOL 'Welcome :)

"Cyril" pronounced "see real". I code stuff.

http://Laissez-Faire.Me/Liberty

"To study and not think is a waste. To think and not study is dangerous." -- Confucius