I have been wondering what is the number of solutions to $a+b=1$ in an arbitrary multiplicative subgroup $H$ (order $r$) of some finite field $GF(q)$, where $q=p^n$ is a prime power and $a, b \in H$. If there is no known results for a general $q$, what about the case $q$ is power of $2$? I'm trying to find a general upper bound on the number of solutions, depending on $p,n$ and $r$.
2026-03-26 12:38:32.1774528712
The number of solutions to $a+b=1$ in a multiplicative subgroup of a finite field $GF(q)$
55 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ELEMENTARY-NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- How do I show that if $\boldsymbol{a_1 a_2 a_3\cdots a_n \mid k}$ then each variable divides $\boldsymbol k $?
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- Algebra Proof including relative primes.
- How do I show that any natural number of this expression is a natural linear combination?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
- algebraic integers of $x^4 -10x^2 +1$
- What exactly is the definition of Carmichael numbers?
- Number of divisors 888,888.
Related Questions in FINITE-FIELDS
- Covering vector space over finite field by subspaces
- Reciprocal divisibility of equally valued polynomials over a field
- Solving overdetermined linear systems in GF(2)
- Proof of normal basis theorem for finite fields
- Field $\mathbb{Q}(\alpha)$ with $\alpha=\sqrt[3]7+2i$
- Subfield of a finite field with prime characteristic
- Rank of a Polynomial function over Finite Fields
- Finite fields of order 8 and isomorphism
- Finding bases to GF($2^m$) over GF($2$)
- How to arrange $p-1$ non-zero elements into $A$ groups of $B$ where $p$ is a prime number
Related Questions in ABELIAN-GROUPS
- How to construct a group whose "size" grows between polynomially and exponentially.
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- Invariant factor decomposition of quotient group of two subgroups of $\mathbb{Z}^n$.
- Computing Pontryagin Duals
- Determine the rank and the elementary divisors of each of the following groups.
- existence of subgroups of finite abelian groups
- Theorem of structure for abelian groups
- In the category of abelian groups the coequalizer $\text{Coker}(f, 0)$, $f: A \to B$ is simply $B/f(A)$.
- Commutator subgroup and simple groups
- Are there any interesting examples of functions on Abelian groups that are not homomorphisms?
Related Questions in CYCLOTOMIC-FIELDS
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- How to write cyclotomic field extensions $\mathbb{Q}(\zeta_n)$ in terms of roots and $i$?
- Question on a proof related to Iwasawa's theorem in Washington's "Introduction to cyclotomic fields"
- Prove that tan(Pi/n)/tan(kPi/n) is an algebraic integer when gcd(k,n) = 1
- Galois group of $x^n+1$ over $\Bbb Q$
- $\mathbb{Q}(3^{1/n},e^{2i\pi/n})$ is not contained in any cyclotomic extension
- How can I prove that the cyclotomic integers $\frac{\zeta_p^r - 1}{\zeta_p^s - 1}$, with $\ p\nmid rs$, are units?
- Determine Minimal Polynomial of Primitive 10th Root of Unity
- Generalization of $x^n-1$ following Fermat's Little theorem
- How to program formula involving cyclotomic polynomials and Lambert series?
Related Questions in ADDITIVE-COMBINATORICS
- Exercise 1.1.6 in Additive Combinatorics
- Show that $A+B$ contains at least $m+n-1$ elements.
- Advantage of Fourier transform on $\mathbb{Z}_N$
- Sorting on non-additive ratios
- Asymptotic formula for the integral sequence s(n)
- Show that $|A+A| < 2.5 |A| $ with $A = \{ [n \sqrt{2}] : 1 \leq n \leq N \}$
- show that $[n \sqrt{3}]$ is an approximate group
- A combinatoric solution (closed expression) for $\sum_{k=i}^n \binom{n}{k}p^k(1-p)^{n-k}$
- On Gowers' approach of Green-Tao Theorem ($\mathcal{D}f$s span $L^q(\mathbb{Z}_N)$).
- Is that specific function additive under disjoint union?
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?