jcla1

Spoiler: there is a simple argument against the existence of such a lower bound.

Interesting follow-up question: What is the distance between the set of harmonic numbers and the integers? i.e. is there a lower bound on the difference between a given integer and its closest harmonic number? If so, for which integer is this achieved?

Even today it is exceedingly rare to find a still-well-conditioned bottle of wine that has the capability to have aged for 117 years or so. Most often sweet wines are capable of this.

FWIW I was also sceptical, but just tried it from my phone network and it seems indeed blocked. Wouldn't be the first case of different ISPs using different block-lists. c.f. bs.to

Measure its volume by dunking it in a jug of water and compare its weight with its expected weight (depending on the density of gold).

There is a famous tale of Archimedes doing exactly this when posed with the problem of determining if a certain crown is made of pure gold.

Do not underestimate the urge to procrastinate (by still doing productive things, like learning Mandarin) while pursuing a PhD.

I am not sure if this will be the author's experience too, but pursuing a PhD will often leave you exhausted without any hope of ever finding "the final missing ingredient" to solve the problem you are currently tackling. So turning to entirely unrelated problems, however productive they may seem to outsides, suddenly becomes an attractive alternative in order to procrastinate.

You can get gcc to generate the blend instruction in the example given - you just have to 'force' it to use SIMD more aggressively by, for example, using -march=native.

I recon the general cosensus among mathematicians (as that is what counts) is that the ABC conjecture so far has _not_ been proven. Mochizuki (and his school around him) seem to be the majority of people that believe his proof is correct. As you point out, Scholze has identified a supposed flaw in Mochizuki's argument, but anyone not already at the forefront of IUT/NT/ABC conjecture is probably incapable of telling if this flaw is a true flaw or not. As Mochizuki refuses to elaborate (on this supposed flaw) consensus cannot be reached and thus the ABC conjecture remains open.

I think giving any more details than his three short posts would delve too far into the specific arguments of the paper to be valuable for a wide enough audience, subtracting those that are going to be looking at the pre-print anyhow.

Unfortunately the presented method only works for the most simple linear recurrence relations.

Tangentially: look up the master theorem if you're interested in at least estimating growth rates for recurrence relations that crop up in computer science.

Just because of the impossibility of an exact formula for the roots of a high-degree polynomial that does not rule out the possibility of figuring out, say, the distribution of roots of such polynomials. The question is not about any polynomial in particular (hence Abel's theorem is no barrier).

edit: Think of the following example: take a polynomial a_n x^n + ... + a_0, where the coefficients a_i are i.i.d. Bernoulli random variables. Even though the degree n might be large (> 4) I can say with confidence that such a polynomial has a real root (x = 0) with probability 1/2. Similar though more sophisticated arguments are at work in the linked question.

I do (I'm a mathematican). We are usually between 4 and 10 people marking an exam with anywhere between 50 and 600 participants.