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New answers tagged nt.number-theory

• 121

What actually is the idea behind the condensed mathematics?

I don't pretend to have anything more than a superficial understanding of condensed mathematics, but Scholze's lecture notes (on condensed mathematics and analytic geometry) are so clearly written ...
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Does the discriminant of an irreducible polynomial of a fixed degree determine the discriminant of the number field it generates?

The discriminants of the irreducible polynomials $$(x^2-2)^2+60 = x^4 - 4 x^2 + 64, \qquad (x^2+2)^2+60 = x^4 + 4 x^2 + 64$$ are both equal to $58982400 = 2^{18} \cdot 3^2 \cdot 5^2$. However, the ...
• 161

A diophantine equation inspired in a conjecture due to Gica and Luca, example of a large Mersenne exponent

The equation (1) for a fixed $x$ is equivalent to the congruence: $$y^2 \equiv 2(1+z^2)\pmod{x}.$$ For $x=25964951$, we have $2\equiv 3328351^2\pmod{x}$, and thus all solutions are obtained from those ...

Priming for the primes

This concerns prime powers and not strictly primes, but the fact that the topological Tverberg conjecture (see e.g. https://arxiv.org/abs/1605.05141) holds for prime power values of “r” and not any ...

Priming for the primes

Virtually all important examples of error-correcting codes that have a chance to be optimal and practical in applications utilize prime numbers.
1 vote

Priming for the primes

Prime numbers occur in a major way in the theory of tactical configurations, in particular in finite geometries.

class number and negative Pell equation

Added: my indefinite forms are reduced in the sense of Gauss and Lagrange. That is, $\langle a, b, c \rangle$ means the form $f(x,y) = a x^2 + b xy + c y^2$ with discriminant $\Delta = b^2 - 4ac.$ ...
• 24.5k

Priming for the primes

Feedback with carry shift register sequences (FCSRs) are an arithmetic parallel of the LFSRs in the answer by @WlodAA. They can be represented using $N-$adic numbers and achieve maximal period when $N$...

Priming for the primes

Here is a characterization of entropy functions due to Faddeev in 1956 (see pp. 229-231 of Faddeev's paper here if you read Russian or Chapter 1 of A. Feinstein's 1958 book Foundations of information ...

Priming for the primes

For me, the most spectacular fact that uses primes in an essential way and which no prime number theorist would be interested in is the fact that Tarski monsters exist for all sufficiently large ...

Priming for the primes

(Very large) prime numbers are of particular importance in cryptography. See e.g. RSA algorithm. I remember also some quasi random number generators using prime numbers.

Priming for the primes

Primes even appear outside mathematics, here is an example from biology: The fact that some species of cicadas appear every 7, 13, or 17 years and that these periods are prime numbers has been ...

Priming for the primes

Regular polygons which you can construct with compass and straightedge have $2^k \cdot p_1 \cdot p_2 \dots p_n$ edges, where $p_i$ are all distinct Fermat primes (so not all primes). It's the reason ...

Priming for the primes

There are maximal shift registers of length $\ p^k-1\$ where $\ p\$ is an arbitrary prime, and $\ k\$ is an arbitrary natural number.

Priming for the primes

Obvious, but still worth mentionning, because the trick is frequently helpful: to represent a collection of elements of different types, where you can have more than one element per type, and order ...

Priming for the primes

The nonzero characteristics of fields are precisely the prime numbers.

Priming for the primes

The abelian simple groups are precisely the groups of prime order.

Generalization of the geometric series representation of the Kronecker delta for arbitrary lattices

The approach indicated by @AlexanderKalmynin is a good heuristic, for sure. Still, there is a somewhat larger context that may be more explanatory, depending on one's tastes. Namely, we can see that ...
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• 6,388

Computing Thompson series for the monster group

In the OEIS index, see "McKay-Thompson sequences or series for Monster simple group, sequences related to" These may come with tables of coefficients, and with programs to compute ...
• 37.2k

A question about generalized harmonic numbers modulo $p$

Glaisher's I-numbers are described in J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., 31 (1899), 216-235.
• 13.3k
1 vote

Understanding inequality in "Small gaps between primes"

For your questions it suffices to study the inner sum $$\sum_{\substack{u_1,....,u_k < R \\ p\mid u_i,u_j \\ (u_i,W)=1 \forall i}}\prod_{i=1}^k \frac{\mu(u_i)}{\varphi(u_i)}.$$ As was pointed out ...

The origin of the Ramanujan's $\pi^4\approx 2143/22$ identity

FWIW, I found a source according to which He found this by first squaring the square of π which gives 97.40909103…, then by subtracting 9^2 = 81 he got 16.40909103…, multiplying this by 22 he got 361....
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Accepted

• 4,578
Accepted

Divergence of primes dividing polynomials

Yes, the series diverges. We can reduce easily to the case of irreducible monic $Q$. Next, let $\alpha_Q(p)$ be the number of roots of $Q(x)$ in $\mathbb{Z}/p\mathbb{Z}$. Note that $M_Q$ is the set of ...
• 9,039
1 vote

It can't be done. By the MRDP Theorem, diophantine sets are the same as c.e. (or Turing Recognizable) sets, and therefore the problem of determining whether two diophantine equations $\varphi, \hat\... • 722 1 vote On the Hilbert function of a numerical semigroup Let$\text{e}$be the minimum of$S^*$. Call a height function a function from$\mathbb Z/\text{e} \mathbb Z$to the natural numbers. The height function$A$for the numerical semigroup$S^*\$ is ...
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