# Questions tagged [number-theory]

Questions on advanced topics - beyond those in typical introductory courses: higher degree algebraic number and function fields, Diophantine equations, geometry of numbers / lattices, quadratic forms, discontinuous groups and and automorphic forms, Diophantine approximation, transcendental numbers, elliptic curves and arithmetic algebraic geometry, exponential and character sums, Zeta and L-functions, multiplicative and additive number theory, etc.

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### Harmonic Distribution of prime numbers

I developed a sieve that depicts the distribution of prime numbers as contained in harmonic (repetitive) patterns. Published it here What would be the process to know if I’m rightfully thinking this ...
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### Questions about Hooleys Approach in Artin's primitive root Conjecture

I know that I have asked 5 questions but they are all part of same proof. To each answer that answer 3 or more questions I will grant a bounty of 100 points and if someone answers all 5 questions I ...
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### What are the fixed points of the arithmetic derivative over the non-negative integers?

I just watched When the derivative of a number is not zero -- The arithmetic derivative by Michael Penn. I like to explore things visually and computationally, so I found this recursive implementation ...
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### Perfect squares can't be primitive roots [duplicate]

This question was asked in my assignment on number theory and I am struck on this. Question: Prove that perfect squares can't be primitive roots if p>2. Attempt: let a is a perfect square and on ...
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### Primitive root and prime p such that p' =4p +1 is also a prime

The following question is from my assignment in number theory and I am not able to make any progress on this. Question: If p is a prime of the form p=4p'+1 where p' is also a prime then 2 is a ...
1 vote
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### 2 questions in the theory of Counting Perfect Squares

I have been reading sieve theory from notes of zeev rudnick here :http://www.math.tau.ac.il/~rudnick/courses/sieves2015.html and in page 2 of lecture 16(http://www.math.tau.ac.il/~rudnick/courses/...
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### Show that an analytic function is always negative

I want to show the following function is negative for $z\in [0,1)$: $$f(z) = -1 + z^2(z-1) + 2\sum_{k=0}^\infty (-1)^k z^{(2k+1)^2+1}.$$ By Tauberian theorem, I know that $\lim_{z\to 1^-}f(z)=0$. We ...
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### Is there a closed-form expression for $f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))$ assuming $x\in\mathbb{R}$?

Question: Is there a closed-form expression for $$f(x)=\sum\limits_{k=1}^\infty (\text{Ei}(i k x)+\text{Ei}(-i k x))\,,\quad x\in\mathbb{R}\tag{1}$$ where $\text{Ei}(z)$ is the exponential integral ...
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### Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
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### Binary quadratic forms over Z of discriminant D [closed]

For each of the following 3 discriminant D = 4d, how can I classify all binary quadratic forms over Z of discriminant D. Is there a way to describe the river corresponding to the canonical form Q(x, y)...
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### Number Theory , Primes and sum of squares [duplicate]

Show that if $a^2+b^2≡0 \pmod{p}$ , with $p$ a prime number and $p≡3\pmod{4}$ Then automatically $a≡0\pmod{p}$ and $b≡0\pmod{p}$ What I have done so far, Suppose $a^2≡0\pmod{p}$ and $b^2≡0\pmod{p}$ , ...
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Find all equivalence classes of positive definite binary quadratic forms over $\mathbb{Z}$ with discriminant $D = −164$ under the action of the group $SL_2(\mathbb{Z})$. Can you find the ...