Testwiki:Reference desk/Archives/Mathematics/2024 August 31

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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)

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

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)

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)

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)

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

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

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)

The prime component? $3240$ has three mpp factors: $2^3,3^4$ and $5^1.$  --Lambiam 12:58, 2 September 2024 (UTC)

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

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

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