Wikipedia:Reference desk/Archives/Mathematics/2009 April 16

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April 16

Prime number

Which is the biggest prime number? —Preceding unsigned comment added by 59.92.243.47 (talk) 11:16, 16 April 2009 (UTC)

See Prime number#The number of prime numbers. —JAO • T • C 11:18, 16 April 2009 (UTC)

Or if you're interested in the largest known prime number, try (surprise!) Largest known prime number. —JAO • T • C 11:21, 16 April 2009 (UTC)

There can be no largest prime number. If there were, list them- {p₁,..., p_k}. Now the product of all the prime numbers in this list is divisible by every prime in this list. Therefore, this product +1 can be divisible by none of the primes in the list (the reminder will be 1 if you try to divide it by any prime). Therefore, This product +1 must be a prime must have a prime factor. This prime factor cannot be any of the primes listed, for otherwise it would divide 1 - a contradiction; we specifically assumed those p_k's to be the only primes. Therefore, they cannot be finitely, but, rather, infinitely many primes. Therefore, there is no largest prime number. If you are interested in a topological proof of this fact, try Furstenberg's proof of the infinitude of primes. The proof I gave is due to Euclid. --PST 12:23, 16 April 2009 (UTC)

The proof you gave isn't quite Euclid's, even though many respectable authors erroneously report that it is. Euclid just said: suppose you have any finite set of primes (not assumed in advance to contain all primes and not assumed to be the first n primes for some n), then multiply the and add one, and look at the prime factors of the number you get. Whether that number is itself prime or not, its prime factors cannot be in the set you started with. Therefore any finite set of primes can be extended to a larger finite set of primes. Making it into a proof by contradiction just copmlicates the proof needlessly and can lead to confusions such as that expressed by McKay below. Michael Hardy (talk) 18:50, 22 April 2009 (UTC)

Ummm... The product+1 must have a prime factor other than the ones you started with. It doesn't need to be prime itself. 2*3*5*7*11*13+1 = 59*509. McKay (talk) 12:52, 16 April 2009 (UTC)   On second thoughts, it should be said like this: (1) Every integer > 2 must have a prime factor. (2) If there are finitely many primes, their product + 1 contradicts (1). McKay (talk) 13:09, 16 April 2009 (UTC)

No - I am not convinced. My proof assumed the hypothetical situation that there are only finitely many primes with a largest prime (say) p_k. The product of all these primes +1 must be larger than all these prime numbers. Furthermore, it must be prime, because by assumption the only primes are those p_j's, and if it were not prime, it would have to be divisible by some p_j. Your counterexample does not demonstrate a fallacy in the proof, because we conclude the product + 1 is prime, because we have already listed all the primes (so it cannot contain a prime factor "not in this list"). Anyway, I may well be wrong - I'm a bit sleepy. :) --PST 14:07, 16 April 2009 (UTC)

I am wrong. I have corrected my (embarrasing) mistake - thankyou for the correction. --PST 14:29, 16 April 2009 (UTC)

However your previous proof by contradiction seems perfectly correct to me.--pma (talk) 15:15, 16 April 2009 (UTC)

Well, I think, OP wanted to ask the largest known prime number so far, which is 2^43,112,609 − 1, a Mersenne Prime. The new largest prime numbers discovered are almost always Mersenne primes because of the algorithmic and computational ease to verify their primaliy. - DSachan (talk) 15:41, 16 April 2009 (UTC)

Really? I always thought it was because Mersenne numbers (with a prime n) were much more likely to be prime then randomly selecting another number... But it looks like you are refering to some of the Mersenne specific primality test such as the Lucas-Lehmer test for Mersenne numbers. I stand corrected. Anythingapplied (talk) 16:32, 16 April 2009 (UTC)

That's also a reason. That's why GIMPS exists and the existence of Lucas-Lehmer test helps their cause. - DSachan (talk) 10:19, 17 April 2009 (UTC)

Not to sound grumpy, but don't the articles I linked to cover pretty much all of this? —JAO • T • C 16:33, 16 April 2009 (UTC)

Yes, they do. Did you expect people to bother with reading what other people wrote before posting? — Emil J. 16:59, 16 April 2009 (UTC)

Right, but then you too shouldn't have written that sentence, but only put a link to it, for it has been repeated soo many times... ;-) pma (talk) 17:52, 16 April 2009 (UTC)

I hate those "...3 football fields long" or "...to the moon an back 11 times" comparisons but I can't help it, how many pages would it take to write out 2^43,112,609 − 1? -hydnjo (talk) 22:44, 16 April 2009 (UTC)

It has 12978189 decimal digits. Decide how many digits you can fit on a page and work it out. Algebraist 23:28, 16 April 2009 (UTC)

Right, that's the hard part. I have no clue nor could I find a reliable source as to how many "standard" size characters fit on a "standard" size page. -hydnjo (talk) 23:42, 16 April 2009 (UTC)

This exact issue has been discussed in Wikipedia. Mersenne prime#List of known Mersenne primes says: "To help visualize the size of the 46th known Mersenne prime, it would require 3,461 pages to display the number in base 10 with 75 digits per line and 50 lines per page." Earlier there was an unsourced claim about "a standard word processor layout", but I changed it in [1] after discussion at Talk:Mersenne prime#Word Processor quote. PrimeHunter (talk) 23:51, 16 April 2009 (UTC)

Thanks PrimeHunter, I obviously (embarrassedly) missed that factoid. -hydnjo (talk) 00:15, 17 April 2009 (UTC)

Oooh, a pile of pages 25 inches high! -hydnjo (talk) 00:29, 17 April 2009 (UTC)

water disturbance equation

Is there a mathematical equation that would describe the amount of time it takes for ripples to stop after a stone is dropped into a nice flat pond if we can assume there are no further disturbances? 65.121.141.34 (talk) 19:56, 16 April 2009 (UTC)

Suppose you were to throw the pebble at point P on the pond. If there are no disturbances, I would assume the ripples to expand until they reach the edges of the pond. So let R be the minimum of all distances from P to points on the boundary of the pond. Assuming the riples enclose a circular area, they will stop expanding after the circle they enclose has radius R. Now all we need to know is the area of the circular region enclosed by the initial ripple and the rate at which the largest ripple's radius is increasing, or the rate of change of the area enclosed by the ripples. Then using calculus, we could work out the answer. If, assuming a perfect model, you wanted to work out the time it takes for the ripples to stop (i.e hit the edges of the lake), using the trajectory of the pebble (when you threw it into the pond), you would have to use wave mechanics. Often, in real life situations, mathematical equations rarely give you the exact answer. --PST 02:20, 17 April 2009 (UTC)

The ripples do not stop. Energy is conserved. Bo Jacoby (talk) 06:36, 18 April 2009 (UTC).

...which is precisely what I said unless of course the pond is a bounded region, in which case the ripples will stop upon reaching the boundary. --PST 08:18, 18 April 2009 (UTC)

If the pond has a boundary then the ripples will be reflected back rather than stopping. Readro (talk) 21:17, 18 April 2009 (UTC)

Yes, although the intensity of the reflection will be damped by frictional effects of the water's interaction with the shore. The extent of such damping will be determined by the material of the shore (a hard boulder might return most of it, while a sandy spot might swallow up all of the ripples). If you want to be really accurate, you'll also need to account for the frictional losses sustained during wave propagation. They're low, but they exist. (Although for a small system and a short time, they probably can be ignored.) -- 75.42.235.205 (talk) 17:10, 19 April 2009 (UTC)

A mistake on my part. In an ideal situation, therefore, the ripples will never cease to be in motion. --PST 03:30, 20 April 2009 (UTC)

PST did you really mean to invoke Schrödinger equation (wave mechanics) here ? Cuddlyable3 (talk) 15:45, 21 April 2009 (UTC)