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prime dilemma


moth
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For any prime number n > 3, n mod 6 = 1 or 5. any prime number n > 3, n mod 3 = 1 or 2. The same prime numbers are in column 1 either way and the primes from column 2 (mod 3) are in column 5 (mod 6).
Are there 2 kinds of prime numbers? Is there a name for these primes?

The attached png is the primes mod 2,3,6, and 7.

primefolds.png

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38 minutes ago, moth said:

For any prime number n > 3, n mod 6 = 1 or 5. any prime number n > 3, n mod 3 = 1 or 2. The same prime numbers are in column 1 either way and the primes from column 2 (mod 3) are in column 5 (mod 6).
Are there 2 kinds of prime numbers? Is there a name for these primes?

I don't think they have names, but there's a famous theorem of Fermat that says an odd prime is the sum of two squares if and only if p = 1 (mod 4). 

https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares

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Thanks for the link. Found some good stuff in the "see also" section too. Now I think I'm misusing the term "Mod". The '%' operator in C is the remainder from integer division, is the result of that operation the same as the mod operation ?

 

 

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58 minutes ago, moth said:

Thanks for the link. Found some good stuff in the "see also" section too. Now I think I'm misusing the term "Mod". The '%' operator in C is the remainder from integer division, is the result of that operation the same as the mod operation ?

Great question. Yes they are "the same but a little different." 

In math, mod is an equivalence relation on the integers. We say that [math]a \equiv b \pmod n [/math] for two integers [math]a, b[/math] if it happens to be the case that [math]n | a - b[/math], where the vertical bar means "divides." So for example [math]5 \equiv 3 \pmod 2[/math] and [math]17 \equiv 5 \mod 3[/math].

You can verify that this is an equivalence relation: It's reflexive, symmetric, and transitive. It partitions the integers into [math]n[/math] pairwise-disjoint equivalence classes. It's a fundamental concept in number theory.

In programming languages, mod is a binary operator that inputs two integers and outputs a third: 5 % 3 = 2. It inputs 5 and 3, and outputs the remainder when 5 is integer-divided by 3.

The math and programming concepts are closely related. [math]a % n[/math] is the remainder when [math]a[/math] is integer-divided by [math]n[/math]; that is, the result of [math]a % n[/math] is the smallest positive element of the equivalence class of [math]a[/math] under the [math]\pmod n[/math] equivalence relation. This turns out to be the same as the remainder under integer division.

The tl;dr is that in math, mod is an equivalence relation that inputs a pair of numbers and a modulus and returns True or False; whereas in programming, mod is a binary operator that inputs an integer and a modulus and outputs the smallest positive member of the equivalence class of the integer mod the modulus.

The difference is shown by, say, the fact that [math]17 \equiv 14 \pmod 3[/math]; but [math]17 \ \% \ 3 = 2[/math] and not [math]14[/math].

That is. even though mathematically [math]17 \equiv 14 \pmod 3[/math], in a programming language, [math]17 \ \% \ 3 == 14[/math] would evaluate to False. That's an example that illustrates the difference.

Edited by wtf
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I think I see the difference now. 5 and 3 (and all odd numbers?) are equivalent mod 2 so the mod operator returns 'true' while the '%' operator returns the same value (1) for (any odd number) mod 2, but the '%' operator would take a few iterations to determine if two integers are in the same equivalence set?

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51 minutes ago, moth said:

I think I see the difference now. 5 and 3 (and all odd numbers?) are equivalent mod 2 so the mod operator returns 'true' while the '%' operator returns the same value (1) for (any odd number) mod 2,

Yes exactly.

 

51 minutes ago, moth said:

but the '%' operator would take a few iterations to determine if two integers are in the same equivalence set?

I suppose you'd have to subtract the two integers and see if the result is integer-divisible by the modulus. That seems like the most sensible way. Probably not the only way.

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Thanks for clearing that up. @wtf.  Reading the Wikipedia page about equivalence classes now. In the pdf i attached, each column is an equivalence class for mod 2, 3, 6, and 5. 

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