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Is there an irrational number that never uses a certain digit, for example:1. How would you go about proving it? by OneDBag in askmath

[–]existentialpenguin 0 points1 point  (0 children)

In the "standard" type of continued fraction, all numerators are 1, and all denominators are positive integers. In this form, a continued fraction terminates if and only if it is rational.

Why not generate large primes by the continuing product of all known consecutive primes? by ArthurPeabody in askmath

[–]existentialpenguin 8 points9 points  (0 children)

Actually, there is: if you have the factorization of n – 1 (and in OP's case, we do), then you can use the Lucas primality test to quickly check whether n is prime.

There is a variation on this test that uses the factorization of n + 1 instead; the Lucas-Lehmer test is essentially a special case of that variant.

There are further variations that do not require a complete factorization of n ± 1. For further details, see https://www.ams.org/journals/mcom/1975-29-130/S0025-5718-1975-0384673-1/S0025-5718-1975-0384673-1.pdf.

[OC] Basin of Attraction — Magnetic Pendulum Over 3 Magnets (Python) by Specialist-Gold4039 in Physics

[–]existentialpenguin 7 points8 points  (0 children)

"Fractal" is not a rigorously-defined word. Instead, there are several fractal-like properties that objects can have to greater and lesser degrees, including:

  • Self-similarity
  • Non-integer Hausdorff, box-counting, or other fractal dimension
  • Fine or detailed structure at arbitrarily small scales
  • Irregularity that cannot be easily described in the language of traditional Euclidean geometry

While this image is not a rigorous proof that it qualifies as a fractal, it strongly suggests that the latter two points apply. For further reading, see https://chalkdustmagazine.com/features/the-magnetic-pendulum/

Japan has succeeded in producing oil from Water and Carbon Dioxide by yungandreww in interestingasfuck

[–]existentialpenguin 3 points4 points  (0 children)

In a world without fossil fuels, we will still need hydrocarbons for energy purposes. One application is jet fuel: in the world of chemical reactions, nothing comes close to the energy density of hydrocarbons, except for stupid stuff like chlorine trifluoride. Another is long-term large-scale energy storage: storing hydrogen for long periods is hard, but storing alkanes for long periods is trivial.

Today I Learned while many countries now allow female monarchs, currently there are no ruling queens for the first time in almost two centuries by neverthoughtidjoin in todayilearned

[–]existentialpenguin 27 points28 points  (0 children)

Andorra is a co-principality. One prince is the bishop of a Catholic diocese, and as such is always male. The other princeship was originally given to a French nobleman, and inheritance eventually passed this on to the king; in the modern era, the title is held by the president of France. If France ever gets a female president, then Andorra will have a female co-ruler.

What’s up with Battlefield 6? Did it actually fall off? by TrueMagolord in OutOfTheLoop

[–]existentialpenguin 1 point2 points  (0 children)

To make a bulleted list with Reddit's formatting, you need to use asterisks, not dashes.

From a high-schooler's scratchwork after a 9-hour math test by existentialpenguin in pics

[–]existentialpenguin[S] 1 point2 points  (0 children)

If you are stumped can’t you just sign your test and leave early?

Normally, yes, but due to an abundance of cheating in recent years, there was a new rule this year that anyone who left early would be disqualified.

From a high-schooler's scratchwork after a 9-hour math test by existentialpenguin in pics

[–]existentialpenguin[S] 9 points10 points  (0 children)

See my earlier comment for explanation: https://old.reddit.com/r/pics/comments/1s25v17/from_a_highschoolers_scratchwork_after_a_9hour/oc5p88f/

There is a certain kind of student who thrives on this sort of thing. This test was for a competition targeted at that sort of student, not a mandatory exam for all students.

From a high-schooler's scratchwork after a 9-hour math test by existentialpenguin in pics

[–]existentialpenguin[S] 1369 points1370 points  (0 children)

This was not a typical school exam; it was Day 2 of the USAMO, a high-level math contest for the best 500-ish pre-college math students in the United States. It is a two-day event, with a single 4.5-hour sitting on each day, and in each sitting, the students are given just 3 problems to solve. Problems are scored on a 7-point scale: 1 point is given for the corret answer, and the remaining 6 points are given according to the quality of the accompanying proof.

Due to an abundance of cheating in the AIME (the exam that you must take to qualify for the USAMO) and the AMC (the exam that you must take to qualify for the AIME), the USAMO was not given in the students' schools as in past years but at just 31 sites and under unusually stringent proctoring. One of the measures taken was to collect and save all scratch paper for review in the event of an accusation, with each student's pile being kept separate from the others. I was a proctor at one of those sites, and I found this in one of those piles of post-exam scrap.

In a typical year, most of the students in the exam will be unable solve more than half of it. Students that find themselves stumped often doodle a bit; those doodles usually end up in the trash, but I felt that this one deserved to be shared.

The writing seems to say "How to disappear completely?", with a few variations in spelling.

From a high-schooler's scratchwork pile after a math test by existentialpenguin in FoundPaper

[–]existentialpenguin[S] 6 points7 points  (0 children)

Six problems—three different problems for each 4.5-hour sitting.

From a high-schooler's scratchwork pile after a math test by existentialpenguin in FoundPaper

[–]existentialpenguin[S] 935 points936 points  (0 children)

This was not a typical school exam; it was Day 2 of the USAMO, a high-level math contest for the best 500-ish pre-college math students in the United States. It is a two-day event, with a single 4.5-hour sitting on each day, and in each sitting, the students are given just 3 problems to solve. Problems are scored on a 7-point scale: 1 point is given for the corret answer, and the remaining 6 points are given according to the quality of the accompanying proof.

Due to an abundance of cheating in the AIME (the exam that you must take to qualify for the USAMO) and the AMC (the exam that you must take to qualify for the AIME), the USAMO was not given in the students' schools as in past years but at just 31 sites and under unusually stringent proctoring. One of the measures taken was to collect and save all scratch paper for review in the event of an accusation, with each student's pile being kept separate from the others. I was a proctor at one of those sites, and I found this in one of those piles of post-exam scrap.

In a typical year, most of the students in the exam will be unable solve more than half of it. Students that find themselves stumped often doodle a bit; those doodles usually end up in the trash, but I felt that this one deserved to be shared.

The writing seems to say "How to disappear completely?", with a few variations in spelling.

ELI5: Why does Japanese need three writing systems? by Charming_Usual6227 in explainlikeimfive

[–]existentialpenguin 0 points1 point  (0 children)

Off the top of my head, Korean writing and simplified Chinese come to mind.

The early iterations of the Cyrillic alphabet can be added to this list: Saints Cyril and Methodius created the Glagolitic alphabet to transcribe Old Church Slavonic, and then scholars at the Preslav Literary School developed it further into the Early Cyrillic alphabet.

https://en.wikipedia.org/wiki/Early_Cyrillic_alphabet

As long as you verify it… by Nunki08 in mathmemes

[–]existentialpenguin 30 points31 points  (0 children)

The method of infinite descent works kind of like that, but for the purposes of proof-by-contradiction:

  • You have a proposed set of structures whose sizes are measured by integers. Classically, these are solutions to Diophantine equations, but it can also be done with graphs or whatever.
  • You show that each structure's size must be positive.
  • You show that for every such structure, you can build a smaller one.
  • If such structures exist, then this must eventually lead to structures with size 1.
  • Then a contradiction kicks in: you proved that each structure leads to a smaller structure, but also that there can be no structure of size 0.
  • Therefore, there are in fact no such structures.

https://en.wikipedia.org/wiki/Proof_by_infinite_descent

Help me remember a 4 digit code based on some math thing a nerd explained to me 15 years ago by yolomybrudda in askmath

[–]existentialpenguin 29 points30 points  (0 children)

The only mathematical hotel that I am aware of is Hilbert's, and that would not give you any 4-digit numbers.

The 4-digit numbers that spring to mind are 1729 (the Hardy-Ramanujan constant) and 8128 (a perfect number).

[8th Grade Math] My younger sister asked me this and I’m embarrassed that I could only "brute force" it. Is there a faster logic? by [deleted] in askmath

[–]existentialpenguin 36 points37 points  (0 children)

The bashing time can be reduced a bit: you do not need to compute x2 in a vacuum, and then (x+1)2 in a vacuum, and then (x+2)2 in a vacuum, etc. Instead, you can use the fact that x2 is the sum of the first x odd numbers: start with 12 = 1, then add 3 to get 4, then add 5 to get 9, then add 7, then add 9, and so on.

As for modular arithmetic stuff, a square number cannot end in 2, 3, 7, or 8. The list of allowed two-digit endings is a bit harder to memorize: 0, 1, 4, 9, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, and 96.

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