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July 26

Divisibility of units

Are there any SI units that divide evenly by 3 (and any of its multiples)? The only such metric units that come to mind are second, whose superunits are 60 times the previous unit (and thus divide evenly by 3), but second itself and its subunits do not, as well as degree of angle, but it is not an SI unit, the corresponding SI unit being radian, which does not divide evenly by 3. And of units that do not divide evenly, are thirds of these units ever used? --40bus (talk) 21:34, 26 July 2025 (UTC)[reply]

According to our article, minutes are non-SI units, so even the example you gave isn't valid. A minute is said to be "accepted for use" though, and if you look at the corresponding table it says minutes, hours and days as time units, and degrees and arcminutes as angle units, have subdivisions divisible by 3. That's 24 in the case of a day, and 60 for everything else. Astronomers also use light-hours, light-minutes, and light-seconds on occasion, but I gather these aren't strictly SI units. A parsec is based on an arcsecond, but it doesn't scale; 60 parsecs do not make an parmin, and if anything it would be the other way around. (Astronomy seems to be the outlier here since it still uses a plethora of non-metric units while the rest of the scientific community has (mostly) converted to SI units.) Really, the whole point of SI units is to use precise decimals instead of less precise fractions, and to reduce the number of units so there is less need for awkward conversion factors. --RDBury (talk) 23:03, 26 July 2025 (UTC)[reply]

If the list of (prefixless) SI units in International System of Units is complete, the answer is no. The only even non-negative divisors of (prefixed) SI-units are the numbers that can be written in the form

2^{m}5^{n}.

‑‑Lambiam 13:59, 27 July 2025 (UTC)[reply]

Are thirds of metric units ever used? More numbers are divisible by 3 than 5, and every SI unit is divisible by 5. How common is to write fractions with metric units? --40bus (talk) 22:34, 27 July 2025 (UTC)[reply]

Of course.:

Therefore, there was a loss of one third of a liter of electrolyte-free water in each liter of urine.^[1]
Now, in vessel A there is one third of a liter of water and in vessel D there are two and thirds of a liter of wine ready for mixing.^[2]
You administer a third of a liter of normal saline and obtain a right-sided 12-lead ECG.^[3]

For a use of fractions, just look here. Editors of scientific journals will ask authors to use decimal notation that complies with the rules in section 5.4 of The International System of Units.

BTW, this is not a maths question. In future, please post any similar questions either in the language section (if it is about actual use observed in the wild) or the science section (if it is about clarifying the rules).

‑‑Lambiam 03:38, 28 July 2025 (UTC)[reply]

I think that's a bit of a nitpick; in my view this query can be thought of as mathematical as much as it can linguistic or scientific, particularly as the querant is not a native English speaker, and different cultures can view concepts differently. And surely most regular respondents on the Ref desks look at all of them (as I do)?

If it bothers you that much, why don't you transfer it to one of those other desks? Which would you pick? {The poster formerly known as 87.81.230.195} 90.193.253.201 (talk) 12:47, 28 July 2025 (UTC)[reply]

July 27

What’s the difference between commutative local fields and number fields ?

Everything is in the question… As far I understand number fields are Global Fields and thus a local field can’t be at the same time a local field isn’t it ? In addition to this, what are the differences ? 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 16:07, 27 July 2025 (UTC)[reply]

A local field is locally compact and complete with respect to a valuation. In characteristic zero, the real and complex numbers are the only archimedean local fields, with the non-archimedean local fields are finite extensions of the p-adic fields. A number field is a finite algebraic extension of the rationals. (In prime characteristic, the global fields are the function fields, and the local fields are completions of function fields with respect to a valuation.) Local fields are always uncountable, and global fields countable. Tito Omburo (talk) 16:21, 27 July 2025 (UTC)[reply]

In my case I was thinking about the case of lifting elliptic curve to local field since it preserve the discrete logarithm. This was in this way I was thinking to local fields. 2A01:E0A:ACF:90B0:0:0:A03F:E788 (talk) 12:28, 28 July 2025 (UTC)[reply]

For a lift to a local field which is unramified, there are basically two approaches. One approach, where you have the polynomial modulus f defining the finite field, you can just use that polynomial to define the quotient field

\mathbb {Q} _{q}=\mathbb {Q} _{p}[x]/(f(x)).

. In sage, you can do this with the class Qq(q, modulus=f) class (or for the valuation ring Zq(q, modulus=f).) If, on the other hand, multiplication in the field

\mathbb {F} _{q}

is given as a black box, you can use the ring of Witt vectors to define the valuation ring

\mathbb {Z} _{q}

. I don't think sage implements the ring of Witt vectors, but it does implement the required symmetric functions. In the unramified case, there is usually a canonical way to lift an elliptic curve, but when the curve is supersingular, there is no canonical lift. Finally, I'm not sure about the case of lifts of elliptic curves in the case of ramified extensions. Tito Omburo (talk) 13:41, 28 July 2025 (UTC)[reply]

August 2

Aurifeuillian factors

If k is a positive integer which is not perfect power, then k^core(k)-(-1)^floor(core(k)/2) (where core(k) is the squarefree part of k, i.e. OEIS: A007913) always have Aurifeuillian factors, and for squarefree k == 1 mod 4, the left Aurifeuillian factors are OEIS: A230376 and the right Aurifeuillian factors are OEIS: A230378, and for squarefree k == 2, 3 mod 4, the left Aurifeuillian factors are OEIS: A230377 and the right Aurifeuillian factors are OEIS: A230379, but there is still no OEIS sequence for all positive integers k which are not perfect powers, so

Please give the (left and right) Aurifeuillian factors of k^core(k)-(-1)^floor(core(k)/2) (where core(k) is the squarefree part of k, i.e. OEIS: A007913) for all positive integers k<=200 which are not perfect powers
If possible, please give a PARI/GP program that can compute the (left and right) Aurifeuillian factors of k^core(k)-(-1)^floor(core(k)/2) (where core(k) is the squarefree part of k, i.e. OEIS: A007913) for given positive integer k which is not perfect power
For which positive integers k which are not perfect powers, the left Aurifeuillian factors of k^core(k)-(-1)^floor(core(k)/2) (where core(k) is the squarefree part of k, i.e. OEIS: A007913) is prime? Should there be infinitely many such positive integers k which are not perfect powers?
For which positive integers k which are not perfect powers, the right Aurifeuillian factors of k^core(k)-(-1)^floor(core(k)/2) (where core(k) is the squarefree part of k, i.e. OEIS: A007913) is prime? Should there be infinitely many such positive integers k which are not perfect powers?

125.230.10.71 (talk) 02:08, 2 August 2025 (UTC)[reply]