nt.number theory – Where can I find a copy of this article from Chowla and Vijayaraghavan?

Does anyone know where I can find a copy of the newspaper Chowla and Vijayaraghavan, & # 39;& # 39; On the biggest first number splitters & # 39; & # 39; ?

Relevant literature says that this was published in the Journal of the Indian Mathematical Society (1937), but it can not be read from there, to my knowledge.

nt.number theory – How did Gauss find the units of the cubic body $ Q[n^{1/3}]$?

I recently read an article on jstor "Gauss and the early development of algebraic numbers", which describes the genesis of Gauss's ideas on the foundations of the algebraic theory of numbers, among other useful information, he mentions a certain ternary cubic form that Gauss studied in 1808 to try to understand the principles under higher rates of reciprocity (cubic reciprocity in this case).

The particular form is:
$$ F (x, y, z) = x ^ 3 + ny ^ 3 + n ^ 2z ^ 3 – 3nxyz $$ and Gauss attempted to find (rational) solutions to the Diophantine equation $ F (x, y, z) = $ 1. As the article explains, this particular form appears as the norm of the number $ x + vy + v ^ 2z $ (or $ v = n ^ {1/3} $) in the pure cubic field created by joining $ v $ the field of rationals. Since Gauss wanted to know where this expression was equal to 1, this investigation can be interpreted as an attempt to find the units (norm 1 numbers) in this cubic field. Gauss then recorded the units for some values ​​of n and, in some cases, presented the fundamental unit.

I have not found enough information about this Gaussian investigation. So now, to my questions:

  • What was the Gauss procedure? And how does this relate to Gauss's other investigations in the algebraic theory of numbers?

  • Does this have anything to do with the Dirichlet Unity Theorem?? I ask the question because this article says that the Gaussian investigation was "a step in the progression of Lagrange to Dirichlet, the latter having developed in 1842-1846 the general theory of algebraic units …".

Nt.number theory – Sum of two whole squares in arithmetic progressions

Is there an explicit formula in the literature for the number of representations of a positive integer? $ n $ as a sum of two whole squares, the second of which is divisible by $ 5 $? So, that means to count the entire representations of $ n $ by the quadratic form $ x ^ 2 + 5 ^ 2 y ^ 2 $.

I hope that something as beautiful as the formula concerns representations as a sum of two whole squares,
$$ sum _ { substack {d in mathbb {N} \ d text {divide} n}} chi (d), $$ or $ chi $ is the modulo of non-main character $ 4 $.

In general, I would be interested in the number of representations by any quadratic form of the form $ d_1 ^ 2 x ^ 2 + d_2 ^ 2 y ^ 2 $, or $ d_1, d_2 $ are non-zero integers.

nt.number theory – Double sum on lattice points in the circle

A friend asked me the following question:
Evaluate the limit, as $ r rightarrow infty $, of the sum $ displaystyle sum limits _ {(m, n) in C_r} $ $ displaystyle (-1) ^ {m + n} on displaystyle m ^ 2 + n ^ 2 $ where the $ (m, n) $ are the lattice points in the radius of the circle $ r $ centered at 0.
I imagine that we would need a Poisson summation to make an exponential sum and apply an analytic estimate, but I do not find any relevant results. Any help and especially references would be welcome.

Nt.number theory – $ p $ – main twist of an elliptic curve in the cyclotomic $ mathbb {Z} _p $ – extension of a $ p $ -adic field

Let K $ to be a digital field and $ v $ to be a fixed premium above $ p $. Let $ k = K_v $. We have cyclotomics $ mathbb {Z} _p $ extension $ K_ infty / K $ and if $ w $ is a bonus above $ v $ in $ K_ infty $ we write $ k_ infty = K _ { infty, w} $.
Let $ E $ to be an elliptical curve defined on k $ and suppose that he has a good ordinary reduction on $ k_ infty $.

Is there a beautiful explicit description for the $ p $primary torsion points $ E (k_ infty) _ {p ^ infty} $?

Theory nt.number – The maximum number of elements in $ S_n $

Note by $ S_n $ the group of permutations of the set $ {1, ldots, n } $ with composition as a binary operation. Let $ m_n $ indicate the maximum order that an element of $ S_n $ may have. What is the smallest positive integer k $ such as $ lim_ {n to infty} frac {m_n} {n ^ k} < infty $?

nt.number theory – Sufficient condition for the absolute convergence of Fourier series of a function on the quotient adele $ mathbb A_k / k $

Let $ G $ to be a compact abelian group. The unitary characters of $ G $ form an orthonormal basis of $ L ^ 2 (G) $, so each square integrable function $ f: G rightarrow mathbb C $ admits a Fourier expansion

$$ f (x) = sum limit _ { chi in hat {G}} c _ { chi} chi (x) tag {1} $$

where the $ c _ { chi} $ are complex numbers only determined satisfying $ sum limits | c _ { chi} | ^ 2 < infty $, and the right side converges to $ f $ in the $ L ^ 2 $-standard.

If more $$ sum limits | c _ { chi} | < infty tag {2} $$ then (1) is actually a point limit (and actually a uniform limit).

When $ G = mathbb R / mathbb Z $it is well known that a sufficient condition for (2) is that $ f $ to be smooth (even just $ C ^ 1 $).

What is it when $ G = mathbb A_k / k $ for k $ a numeric field, and $ mathbb A_k $ the adeles of k $? There is a notion of smooth function on $ mathbb A_k $ (being smooth in the Archimedean argument and locally constant in the nonarchedian). Does the Fourier series of a smooth function $ f $ sure $ mathbb A_k / k $ to satisfy (2)? If not, is there a sufficient condition well known on $ f $ for (2) to hold?

nt.number theory – New formula for the quadratic field class number $ mathbb Q ( sqrt {(- 1) ^ {(p-1) / 2} p}) $?

I have the following conjecture involving a possible new formula for the class number of the quadratic field $ mathbb Q ( sqrt {(1) ^ {(p-1) / 2} p}) $ with $ p $ a strange bonus.

Conjecture. Let $ p $ to be an odd number and leave $ p ^ * = (- 1) ^ {(p-1) / 2} p $. Then the class number $ h (p ^ *) $ of the quadratic field $ mathbb Q ( sqrt {p ^ *}) $ coincides with the number
$$ frac {( frac {-2} p)} {2 ^ {(p-3) / 2} p ^ {(p-5) / 4}}} det left[cotpifrac{jk}pright]_ {1 le, k le (p-1) / 2}, $$
or $ ( frac { cdot} p) $ is the symbol of Legendre.

This is conjecture 5.1 in my preprint arXiv: 1901.04837. I checked it for all the odd firsts $ p <$ 29. Note that $ h (p ^ *) = $ 1 for each odd premium $ p <23 $, and $ h (-23) = $ 3.

Here, I invite some of you to further test this hypothesis. My computer can not check it even for $ p = $ 29.

nt.number theory – A surprising identity: $ det[cospifrac{jk}n]_ {1 j, k n} = (- 1) ^ { lfloor frac {n + 1} 2 rfloor} (n / 2) ^ {(n-1) / 2} $

On the basis of my calculation, I pose here my next conjecture involving the cosine function.

Conjecture. For any positive integer $ n $we have the identity
$$ frac1 {2n} det left[cospifrac{jk}nright]_ {0 le j, k n} = det left[cospifrac{jk}nright]_ {1 j, k n} = (- 1) ^ { lfloor frac {n + 1} 2 rfloor} (n / 2) ^ {(n-1) / 2}. $$

This is a part of conjecture 5.7 in my pre-print arXiv: 1901.04837. The paper contains more similar conjectures.

Ideas for a solution of the conjecture?

nt.number theory – Vector the shortest of the basic product and its sizing?

Given a lattice $ Lambda $ we can find a base $ v_1, dots, v_n in mathbb R ^ n $ with

$$ | v_i | leq gamma_ {i, n} det ( Lambda) ^ {1 / (n-i + 1)} $$ or $ gamma_i $ is a function only of $ i $ and $ n $.

Organize $ v_1, dots, v_n in mathbb R ^ n $ like rows of a matrix $ M $. So any vector in the networks is of shape $ xM $ or $ x in mathbb Z ^ n $.

  1. If you have another network $ Lambda & # 39; with base $ D $ and the shortest vectors
    $ u_1, dots, u_n in mathbb R ^ n $ with $$ | u_i | leq gamma # {i, n} det ( Lambda & # 39;) ^ {1 / (n-i + 1)} $$ do we know something about lattice $ Lambda & # 39; $ with base $ MD $?

  2. Yes $ D $ is diagonal then $ | v_1 |, dots, | v_n | in mathbb R ^ n $ always provide upper related to the length of the shortest vectors in the network $ Lambda & # 39; $ with base $ MD $?