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August 31

Term for prime to a power?

For the prime factorization of n: $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\cdots p_{k}^{n_{k}}$

is there a term for an individual $p_{i}^{n_{i}}$ ? Bubba73 ^{You talkin' to me?} 04:17, 31 August 2024 (UTC)[reply]

Prime power. AndrewWTaylor (talk) 16:30, 31 August 2024 (UTC)[reply]

I thought of that but it isn't specific enough. What I'm looking for is for the largest power of the prime that divides the number. Bubba73 ^{You talkin' to me?} 20:26, 31 August 2024 (UTC)[reply]

It would be nice if there was a settled answer to this question. "Primary factor" would be appropriate in commutative ring theory. (Primary ideal) But this usage is not standard in this situation. Tito Omburo (talk) 21:44, 31 August 2024 (UTC)[reply]

The exponent of p in the prime factorization is called the p-adic valuation of n. 100.36.106.199 (talk) 01:49, 1 September 2024 (UTC)[reply]

Thanks. Bubba73 ^{You talkin' to me?} 02:28, 2 September 2024 (UTC)[reply]

A bit of a mouthful: "maximal prime-power factor".^[1][2][3] --Lambiam 21:56, 1 September 2024 (UTC)[reply]

Thanks. That is a bit of a mouthful (i.e. too long to use repeatedly). In my mind, and in notes to myself, for years I have called it the "prime component". Soon I expect to be writing up something, so I wondering if there is a recognized term. Bubba73 ^{You talkin' to me?} 02:28, 2 September 2024 (UTC)[reply]

The prime component?

3240

has three mpp factors:

2^{3},3^{4}

and

5^{1}.

--Lambiam 12:58, 2 September 2024 (UTC)[reply]

I'd call each of those a prime component. Bubba73 ^{You talkin' to me?} 21:09, 2 September 2024 (UTC)[reply]

We hereby authorise you to name it the "Bubba factor". -- Jack of Oz ^{[pleasantries]} 20:02, 2 September 2024 (UTC)[reply]

Ha ha. Bubba73 ^{You talkin' to me?} 21:09, 2 September 2024 (UTC)[reply]