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Comment by messe | Hacker News Reader

That's just the first counterexample that was proven to exist. The smallest counter example is less than 10^10 (in fact smaller than 10^9).

> The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3]

> An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest counterexample is n = 906,150,257, found by Minoru Tanaka in 1980.[5]

EDIT: rephrased because despite having a maths degree I can't count.

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I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

I do like to joke that I studied maths because I don't like numbers.

I recall the end of a science fiction novel, wherein we finally meet aliens. It turns out that, as a trend, the species that are good at counting are bad at math and especially counting things at a glance. The species that is best at math have a hard time telling the difference between two things and three things just looking.

I know a young man with dyscalculia and we were hoping when he got to symbolic math that his problems would diminish.

That has not born fruit so far.

I pursued and achieved an EE degree plus a couple extra math courses largely because I was "not good at math" as a youth. Nothing like hearing "I don't think math is your subject" from a key adult to light a fire under ones fanny.

In hindsight I could have had the same career path with a straight CS degree and much less stress during my college years. No regrets though, math really is fun! Even more so when the classroom setting is removed.

When I was about 12yo, I went to the eye doctor with my father (in France). The doctor realized that I had daltonism and told my father in front of me that I would never be an engineer because of that.

At school, we had tests to show what we were good at. The guy who came to comment on our tests told me that I should definitely go for something like literature or history.

And here I am, an engineer with an extra PhD in physics who dreams about meeting these idiots and explaining to them that what they do is revolting. They are probably dead by now though (it was in the 80's)

daltonism : color blindness

Had to look it up.

It's one of the prerequisites in fact.

> I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

Reminds me of a story from <https://www.ams.org/notices/200410/fea-grothendieck-part2.pd...>. Quoting it below:

One striking characteristic of Grothendieck’s mode of thinking is that it seemed to rely so little on examples. This can be seen in the legend of the so-called “Grothendieck prime”. In a mathematical conversation, someone suggested to Grothendieck that they should consider a particular prime number. “You mean an actual number?” Grothendieck asked. The other person replied, yes, an actual prime number. Grothendieck suggested, “All right, take 57.”

This is absolutely my personal experience. I am absolutely awful at arithmetic, but I think pretty competent at decently advanced mathematics.

It's like they say: the only numbers a mathematician needs are 0, 1, and 2 (and just because it's not 1).

In number theory, 2 is often a special case in a lot of theorems. There's something odd about the even prime.

i.e. the first information-encoding base.

I'm not sure. The Fibonacci numbering system has a smaller effective base.

(And there's always unary.)

See https://en.wikipedia.org/wiki/Fibonacci_coding

In analysis and its applications, the L^2 norm is also rather special.

My hot take is that it should be called the L^(1/2) norm. Many theorems and formulas become a lot easier to state if you redefine L^p as L^(1/p).

For instance, under this notation, the dual of L^p is just L^(1-p). And Littlewood’s interpolation inequality is a lot easier to remember, since the exponents used come directly from the coefficients in the convex combination:

If r = ap + bq, where a and b are nonnegative and sum to 1, then |f|_r <= (|f|_p)^a (|f|_q)^b

You should join the folks at the tau-not-pi club.

And going off on a tangent: in thermodynamics, we should measure coldness (coldness ~ 1 / temperature). It makes all the math come out nicer.

See https://en.wikipedia.org/wiki/Coldness

Coldness handles 'negative temperatures' much better. As Wikipedia puts it:

> Though completely equivalent in conceptual content to temperature, β [= coldness] is generally considered a more fundamental quantity than temperature owing to the phenomenon of negative temperature, in which β is continuous as it crosses zero whereas T has a singularity.[7]

“I’m a mathematician, not an arithmetician!”

There are three kinds of mathematicians...

just looked that up, i like.

> EDIT: rephrased because despite having a maths degree I can't count.

You would probably enjoy Tom Körner's book 'The Pleasures of Counting'. See eg https://maa.org/press/maa-reviews/the-pleasures-of-counting

> He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10361

1.845 × 10361 = 19116.045

Thanks, it should be 10^361. I've fixed it.

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