python – how effective is it (Gauss trick to add numbers efficiently)

Function printing the sum of the numbers between 2 given numbers:

def sum_of_numbers (start_number, end_number):

# n explicit
n = abs (end_number - start_number) + 1

# n / 2 represents the number of pairs,
# end_number + start_number
represents the total for each pair.
return (end_number + start_number) * (n / 2)

Function that prints the sum of odd numbers:

def sum_of_odds (start_number, end_number):
"" "start_number <end_number, start_number and end_number are odd" ""
# When adding odd numbers (1 + 3 + 5 + 7 + ....... + 2n-1)
# you will get (1, 4, 9, 16, ......., n ** 2), the total can be represented by
# n ** 2, to find the total of the first, say 100 numbers,
# we use 100/2 ** 2 (half even and half odd) or 2500, it works!

# To make this a little more general, let's say we want to find the odd
# numbers from 5 to 11, it's like adding the whole thing that is 6 ** 2 and
# subtract the first 2 odd numbers (6 ** 2 - 1 - 3)

start_number_position = (start_number + 1) / 2
end_number = (end_number + 1) / 2

return (end_number_position) ** 2 - (start_number_position - 1) ** 2

Prints the sum of even numbers:

def sum_of_evens (start_number, end_number):
"" "start_number < end_number, start_number and end_number are even"""

    # To find even numbers total between 2 numbers, (e.g : 12->20)
# We add the odd numbers before each even number (11 + 13 + 15 + 17 + 19) then we
# add 1 for each number (1 * 5)

return sum_of_odds (start number-1, end-1 number) 
+ (end_number - start_number) / 2 + 1

Posthumous publications of Gauss?

I am looking for information on the posthumous publication of the mathematical correspondence of Gauss and his notebooks.

When did they become widely available and how did that affect progress in mathematics?

reference request – Does the Gauss sum attached to $ chi $ belong to $ mathbb {Q} ( chi) $?

Let $ p $ to be a prime number and $ chi $ to be a primitive character of Dirichlet of conductor $ p $. We leave $$ g ( chi) = sum_ {a = 1} ^ {p} { chi (a) e ^ {2i pi a / p}} $$ to be the sum of Gauss attached to $ chi $. Is it known that $ g ( chi) $ does not belong to $ mathbb {Q} ( chi) $, the algebraic extension of $ mathbb {Q} $ generated by the values ​​of $ chi $?

Thank you so much!

Aggressive geometry – The Gauss lemma for a complete local ring

Assume that $ R $ is a complete Noetherian field on a field K $.

Suppose that a monique polynomial $ f (X) in R[X]$ (that is to say, the highest degree $ X ^ e $ in $ f $ has the coefficient $ 1 $), meets both of the following conditions$ colon $

  1. $ R[X]/ (f (X)) $ is do not integral.

  2. $ f (X) = g (X) ^ l $ in $ F (R)[X]$ for a whole number $ l> $ 1, or $ g (X) $ is an irreducible polynomial in $ F (R)[X]$ and $ F (R) $ is the fractional field of $ R $.

Q. Secondly, is the following equality valid for some people? $ G (X) in R[X]$$ colon $ begin {equation *}
f (X) = G (X) ^ l ~?
end {equation *}

couples of problems related to Gauss flow and vector analysis

the surface $ S $ is given by $ r $ (R,$ theta $) = (r $ cos theta $)$ i $+ (r $ sin theta $)$ j+ $ theta k $. (0 <= r <= a, 0 <=$ theta $<=$ frac { pi} {2} $ ) if the vector feild $ A $ is given by $ A $= x$ i $+ y$ j++ zk $,

1. Get the normal unit vector $ n $ of $ S $

2.évaluer $ int_ {S} A $ cdot n $ and $ int_ {S} left ( bigtriangledown times A right) cdot n dS $

I am new in vector analysis, any index would be appreciated.

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 …".

abstract algebra – On the irreducibility of a polynomial lemma and Gauss

Let $ P (X) = X ^ 5 – 6X + 3 $.

Prove that it is irreducible on $ mathbb {Q} $.

In the solution that I have for this exercise, I literally have:

I miss the necessary knowledge about irreducibility by reducing modulo p. All I know is that there is a bijection between the roots of P in $ overline { mathbb {Q}} $ and the roots of $ overline {P} $ in $ overline { mathbb {F} _p} $.

I also know this generalization of Lemma de Gauss: in a factorial ring A, with its fractional body K, a primitive $ P $ is irreducible in $ A[X] Leftrightarrow $ $ P $ is irreducible in K[X].

How does this apply to P above?

Thank you for any direction or help.