Talk:Prime number

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I struck my own comment as it was useless (due to simple sleight of hand of ref) —Preceding unsigned comment added by Billymac00 (talk • contribs)

Wording re regular polygon construction

My edit has been reverted, restoring the following wording:

A regular n-gon is constructible using straightedge and compass if and only if

n = 2ⁱ · m

where m is a product of any number of distinct Fermat primes and i is any natural number, including zero.

True, the m=1 case is just a null product. But the point of my edit was that the old and restored version claims that (m, n)=(1, 0) or (1, 1) gives the number of sides of a constructible regular polygon: a1-sided polygon and a 2-sided polygon respectively. The former is impossible, and the latter is possible only if we admit a degenerate polygon. So I think the wording needs to be corrected. Loraof (talk) 21:12, 15 March 2016 (UTC)

After David Eppstein dealt with the issue you raised, you have made a second change to make the phrasing more complicated in order to avoid the empty product -- why? (I also am not certain that your new phrasing is correct -- haven't you lost n=3?) Frankly I do not see what the problem with the original wording is -- even in the worst case (if I agree that there is no such thing as a regular 1-gon or 2-gon), we are making a true assertion about an empty set (that all its members have the property of constructability).

On an unrelated note, could someone move the paragraph and image about Fermat primes and constructability into the section on Applications below? It obviously does not belong where it is now. (I am on Amtrak and the connection is not good enough for me to edit the article.) Otherwise I will try to get to it in the next few days. Thanks in advance. --JBL (talk) 02:32, 16 March 2016 (UTC)

My feeling is also that the 2^i m formula is entirely adequate to encompass all cases, using the empty product, and that adding cases to the characterization to avoid the empty product is a mistake (it makes the result harder to understand, not easier). —David Eppstein (talk) 03:27, 16 March 2016 (UTC)

Euler's proof

This article and Euclid's theorem attribute different proofs of the infinitude of primes to Euler. This one says it's about the divergence of the series of prime reciprocals, while the other one involves an Euler product formula for the harmonic series. Are they both Euler's? Shouldn't we be more consistent? —David Eppstein (talk) 21:48, 17 March 2016 (UTC)

I think there is only one proof, of the divergence of the series of prime reciprocals, which makes use of the product formula for the harmonic series: it is contained in "Variae observationes circa series infinites" [1]. Euler establishes through a product formula that the harmonic series is the (hyperbolic) logarithm of infinity ("absolutus infinitus", which he considers as a somehow special number) in Theorema 7. Then he uses the latter to prove in Theorema 19 that the sum of the reciprocal of primes is (hyperbolic) log log of the infinity. Sapphorain (talk) 23:31, 17 March 2016 (UTC)

They are closely related but I'm not convinced that they are really the same proof. To go from divergence of harmonic series to divergence of reciprocal prime series, one uses the product formula, takes the log of the resultiing product, and then argues that the error terms in the log are inconsequential. But the proof at Euclid's theorem stops after using the product formula, arguing that the finitude of primes would already make the product finite, without bothering to take its log. So there's a disconnect between what we say at that article (which doesn't mention the reciprocal prime series) and what we say here (which doesn't mention the harmonic series). —David Eppstein (talk) 18:43, 18 March 2016 (UTC)

Yes, you are right. Euler already mentions as a consequence of his Theorem 7 the infinity of primes, in the Corollary 2 page 174 (in which he remarks they are more numerous than squares): "…sequitur infinities plures esse numeros primos, quam quadratos…" Thus after his Theorem 19 he has of course no reason to repeat that there is an infinity of primes. So the two references to Euler you mention are related to the same proof, but only the assertion that Euler obtained a new proof of Euclid's theorem after proving the product formula (Theorem 7) seems to be appropriate; the assertion that he obtained it after proving the infinity of the sum of reciprocal of primes (Theorem 19) appears to be irrelevant, and to have been popularized by people who (like me) didn't read Euler's proof in details. Sapphorain (talk) 00:07, 19 March 2016 (UTC)

Semi-protected edit request on 2 June 2016

Number of prime numbers

In the list of prime number you have 101 as a prime number, this is incorrect.

Prime Numbers Set Definition

Definition of prime numbers can be written all over as this: Prime numbers does not have a factor other than itself and 1; as following;

For $N\in Z^{+}$;

$a_{i}$ $i^{th}$ row value, $b_{j}$ $j^{th}$ column value which $i,j=1,2,3,\dots N.$

$a_{i},b_{j}=1,2,3,\dots N.$

$x_{ij}=a_{i}*b_{j}$

Sets consists of $x_{ij}$'s are;

$A=\{x_{ij};\quad i,j\in Z^{+}\}$

$B=\{x_{ij};\quad i,j\in Z^{+}-\{1\}\}$

Thus;

$A=\{u_{1},u_{2},u_{3},\dots \}$

$B=\{v_{1},v_{2},v_{3},\dots \}$

P shows prime numbers' set,

$P=A-B-\{1\}$

Some features of sets are like below,

$s(A)>s(B).$

$B\subset A.$

$A=B\cup P\cup \{1\}.$

Set A consists of positive integers starting from 1, set B consists of positive integers greater than 1. Separately multiplying their own members with their own thus getting a new pair of sets. The difference of the new sets is the set of almost prime numbers. 1 is not a member of the prime numbers' set, prime numbers' set could be obtained by substracting 1 from latest found set. If it's desired a N value could be selected by that prime numbers' set can be obtained within the range of 1 to N --Nexusiot (talk) 20:46, 2 June 2016 (UTC) Comment : "Please add this new feature" --Nexusiot (talk) 20:46, 2 June 2016 (UTC)

Nexusiot (talk) 20:46, 2 June 2016 (UTC)

Not done: For several reasons.

1. Your account will be autoconfirmed in about 3 days, at which time, you will be able to make the edit.
2. This request is unclear exactly what needs to be changed or added. It would be easier to process this if you wrote this in a "Change X to Y" format. — Andy W. (talk · ctb) 21:34, 2 June 2016 (UTC)

Prime number formula

All prime numbers have to follow this formula is: 6n±1 — Preceding unsigned comment added by 212.253.111.210 (talk) 00:33, 5 October 2016 (UTC)

Best regards Nedim ERDAN — Preceding unsigned comment added by 212.253.111.210 (talk) 00:36, 5 October 2016 (UTC)

2 and 3 don't. Numbers of form 6n±1 is the same as numbers not divisible by 2 or 3 so it clearly includes all primes above 3. PrimeHunter (talk) 00:52, 5 October 2016 (UTC)

This was discussed here in the talk page archive.--♦IanMacM♦ ^{(talk to me)} 15:38, 19 October 2016 (UTC)