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Computational problem used in cryptography
Short integer solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based
Short integer solution problem
Short_integer_solution_problem
Method to solve optimization problems
variables are required to be integers, then the problem is called an integer programming (IP) or integer linear programming (ILP) problem. In contrast to linear
Linear_programming
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be verified in polynomial time, must the problem be solvable in polynomial time
P_versus_NP_problem
Optimization problem in computer science
Learning with errors Short integer solution problem Khot, Subhash (2005). "Hardness of approximating the shortest vector problem in lattices". J. ACM
Lattice_problem
Problem a computer might be able to solve
science, a problem is one that asks for a solution in terms of an algorithm. For example, the problem of factoring "Given a positive integer n, find a
Computational_problem
NP-hard problem in combinatorial optimization
Corporation, who expressed the problem as an integer linear program and developed the cutting plane method for its solution. They wrote what is considered
Travelling_salesman_problem
Computer software bug occurring in 2038
systems. Modern systems and software updates address this problem by using signed 64-bit integers, which will take 292 billion years to overflow—approximately
Year_2038_problem
Decision problem in computer science
sum problem (SSP) is a decision problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and
Subset_sum_problem
Mathematical problem
be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number
Coin_problem
Problem of determining if a Boolean formula could be made true
finite integer k. #SAT, the problem of counting how many variable assignments satisfy a formula, is a counting problem, not a decision problem, and is
Boolean satisfiability problem
Boolean_satisfiability_problem
On solvability of Diophantine equations
Diophantine equation with integer coefficients has a solution in integers. Hilbert's problem is not concerned with finding the solutions. It only asks whether
Hilbert's_tenth_problem
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Problem in combinatorial optimization
Knapsack Problem Archived 14 February 2015 at the Wayback Machine Optimizing Three-Dimensional Bin Packing Knapsack Integer Programming Solution in Python
Knapsack_problem
In mathematics, when is n!+1 a square
Unsolved problem in mathematics Does n ! + 1 = m 2 {\displaystyle n!+1=m^{2}} have integer solutions other than n = 4 , 5 , 7 {\displaystyle n=4,5,7}
Brocard's_problem
Computer arithmetic error
8-bit integer addition of 127 + 1 results in −128, a two's complement of 128). (A solution for this particular problem is to use unsigned integer types
Integer_overflow
Physics problem related to laws of motion and gravity
universal gravitation. Unlike the two-body problem, the three-body problem has no general closed-form analytic solution. The differential equations that govern
Three-body_problem
Optimization problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a
Vehicle_routing_problem
Mathematical problem in number theory
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of
Waring's_problem
Mathematical puzzle
measure any integer amount up to the sum of the volumes. As shown in the previous section, we can construct the solution to the problem from the desired
Water_pouring_puzzle
NP-complete problem in computer science
science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into
Partition_problem
on the traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is
List_of_NP-complete_problems
Sum of inverse squares of natural numbers
1741. The solution to this problem can be used to estimate the probability that two large random numbers are coprime. Two random integers in the range
Basel_problem
Seven mathematical problems with a US$1 million prize for each solution
correct solution, pledged by the Clay Mathematics Institute. The Clay Mathematics Institute officially designated the title Millennium Prize Problem for the
Millennium_Prize_Problems
Choosing the fewest coins to make a given amount of money
of the integer knapsack problem, and has applications wider than just currency. It is also the most common variation of the coin change problem, a general
Change-making_problem
Complexity class
known as the travelling salesman problem—is NP-hard. The subset sum problem is another example: given a set of integers, does any non-empty subset of them
NP-hardness
Mathematical problem in number theory
cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions
Archimedes's_cattle_problem
{\displaystyle A,B,C} must share some prime factor. Brocard's problem: are there any integer solutions to n ! + 1 = m 2 {\displaystyle n!+1=m^{2}} other than
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Mathematical algorithm
Kuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ax + by
Kuṭṭaka
How many integer lattice points there are in a circle
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and
Gauss_circle_problem
Mathematical puzzle
studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such
The_monkey_and_the_coconuts
Initial set of valid possible values
including inequalities, equalities, and integer constraints. This is the initial set of candidate solutions to the problem, before the set of candidates has
Feasible_region
Problem of finding the best feasible solution
economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two
Optimization_problem
Arithmetic operation
the Greatest Unsolved Problem in Mathematics. New York City: Penguin Books. ISBN 978-0-452-28525-5. Weisstein, Eric W. "Integer Division". MathWorld.
Division_(mathematics)
Classical problem in combinatorics
to form an integer solution. The primal-dual algorithm for the set cover problem is an iterative method that constructs feasible solutions to both the
Set_cover_problem
Solving an optimization problem with a quadratic objective function
x will need to take on integer values. This leads to the formulation of a mixed-integer quadratic programming (MIQP) problem. Applications of MIQP include
Quadratic_programming
Mathematical problem in operations research
problem reducible to the knapsack problem. The problem can be formulated as an integer linear programming problem. A paper machine can produce an unlimited
Cutting_stock_problem
Topics referred to by the same term
state SIS (file format), Symbian OS filename extension Short integer solution problem, a problem in lattice-based cryptography Single-instance storage
Sis
Combinatorial optimization problem
polynomial. If the weights are integers, and all weights are at most C (where C>1 is some integer), then the problem can be solved in O ( m n log (
Assignment_problem
Cryptographic algorithm created by Adi Shamir
demonstrated above, which uses integer arithmetic rather than finite field arithmetic, works, there is a security problem: Eve gains information about S
Shamir's_secret_sharing
Optimization technique for solving (mixed) integer linear programs
Such procedures are commonly used to find integer solutions to mixed integer linear programming (MILP) problems, as well as to solve general, not necessarily
Cutting-plane_method
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
2026 release of AI-generated mathematical solutions
breakthroughs. On 8 September 2026, OpenAI claimed a solution to the Navier–Stokes existence and smoothness problem with a smooth external force. However, approximately
OpenAI bulk release of mathematical papers
OpenAI_bulk_release_of_mathematical_papers
Type of permutation of a set of elements
Dn equals the nearest integer to n!/e, where n! denotes the factorial of n and e ≈ 2.718281828... is Euler's number. The problem of counting derangements
Derangement
Mathematical and computational problem
NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition
Bin_packing_problem
Root of a quadratic polynomial with a unit leading coefficient
are integers, i.e. quadratic integers are algebraic integers of degree two. Thus quadratic integers are those complex numbers that are solutions of equations
Quadratic_integer
Theorem in geometric topology
conjecture is the only Millennium Prize Problem with a verified solution. The Poincaré conjecture was a mathematical problem in the field of geometric topology
Poincaré_conjecture
Problem in number theory
sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and
Sums_of_three_cubes
17th-century conjecture proved by Andrew Wiles in 1994
developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers x and y such that their
Fermat's_Last_Theorem
Unsolved problem in mathematics
Unsolved problem in mathematics Is it possible to construct a three-by-three magic square from nine distinct integer squares? More unsolved problems in mathematics
Magic_square_of_squares
23 mathematical problems stated in 1900
Fields Medal in 1966 for his work on the first problem, and the negative solution of the tenth problem in 1970 by Yuri Matiyasevich (completing work by
Hilbert's_problems
Set of objects whose state must satisfy limits
these kinds of problems. Additionally, the Boolean satisfiability problem (SAT), satisfiability modulo theories (SMT), mixed integer programming (MIP)
Constraint satisfaction problem
Constraint_satisfaction_problem
Mathematical counting-out question
used to solve this problem in the general case by performing the first step and then using the solution of the remaining problem. When the index starts
Josephus_problem
Problem in geometry
{n}{4}}+{\binom {n}{2}}+1\right),} equivalent to the given solution to Moser's circle problem. John Horton Conway and Richard K. Guy provide an alternative
Moser's_circle_problem
Complex number whose real and imaginary parts are both integers
number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and
Gaussian_integer
On unit fractions adding to 4/n
{1}{y}}+{\tfrac {1}{z}}} have a positive integer solution for every integer n ≥ 2 {\displaystyle n\geq 2} ? More unsolved problems in mathematics The Erdős–Straus
Erdős–Straus_conjecture
On divisibility among sets of integers
Znám's problem asks which sets of integers have the property that each integer in the set is a proper divisor of the product of the other integers in the
Znám's_problem
Mathematical problem set on a chessboard
queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no
Eight_queens_puzzle
Mathematical riddle
Mathematically, the problem can be formulated as follows: Given an integer m and a set V of positive integers, find the smallest integer z that cannot be
Postage_stamp_problem
Mathematical puzzle
various lengths and heights, or requesting unusual solutions such as cases where all values are integers. Its charm has been attributed to a seeming simplicity
Crossed_ladders_problem
Complexity class
the integers that have the same total. This problem is contained in PPP, but it is not known if it is PPP-complete. The constrained-SIS (short integer solution)
PPP_(complexity)
Family of solutions to related differential equations
when solving problems (like Laplace's equation) in cylindrical coordinates. When α {\displaystyle \alpha } is a half-integer, the solutions are called spherical
Bessel_function
Probability of shared birthdays
Encyclopedia of Integer Sequences. OEIS. Retrieved 17 February 2020. DasGupta, Anirban. "The matching, birthday and the strong birthday problem: a contemporary
Birthday_problem
Concept in integral mathematics
optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed
Linear_programming_relaxation
Triangle with integer side lengths
positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the
Integer_triangle
On the distribution of prime numbers
sometime be in a position to attempt the rigorous solution of Goldbach's problem, viz., whether every integer is expressible as the sum of two positive prime
Hilbert's_eighth_problem
Computational problem of graph theory
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights
Shortest_path_problem
Subfield of mathematical optimization
feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP")
Combinatorial_optimization
Branch of numerical optimization
on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one
Deterministic global optimization
Deterministic_global_optimization
Problem of inverting exponentiation in groups
discrete logarithm problem over carefully chosen groups has no efficient solution. In general, there is no subexponential time solution for black box groups
Discrete_logarithm
Natural number
antidiagonals: T(n,k) = solution to postage stamp problem with n stamps and k denominations (n >= 1, k >= 1))". The On-Line Encyclopedia of Integer Sequences. OEIS
300_(number)
List of unsolved computational problems
list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known or when experts
List of unsolved problems in computer science
List_of_unsolved_problems_in_computer_science
Optimization by removing non-optimal solutions to subproblems
cannot contain the optimal solution. It is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization
Branch_and_bound
Mathematical proof technique
most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it
Vieta_jumping
Unsolved problem about sums of powers
In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials
Prouhet–Tarry–Escott_problem
Problem-solving technique and algorithmic paradigm
satisfies the problem's statement. A brute-force algorithm that finds the divisors of a natural number n would enumerate all integers from 1 to n, and
Brute-force_search
Combinatorial optimization method
maximization problem. The method solves the linear program without the integer constraint using the regular simplex algorithm. When an optimal solution is obtained
Branch_and_cut
Computational problems no algorithm can solve
Hilbert's tenth problem: the problem of deciding whether a Diophantine equation (multivariable polynomial equation) has a solution in integers. For functions
List_of_undecidable_problems
Solution of some Diophantine equation
unknowns y1 and y2. The equation has a solution in y1 and y2 precisely when x can be expressed as a product of two integers greater than 1, in other words x
Diophantine_set
Method in mathematical optimization
difficult problem of constrained optimization by a simpler problem. A solution to the relaxed problem is an approximate solution to the original problem, and
Lagrangian_relaxation
Problem in computational complexity theory
hash table contains the integer − ( S [ i ] + S [ j ] ) {\displaystyle -(S[i]+S[j])} . It is also possible to solve the problem in the same time in a comparison-based
3SUM
Type of computational problem
that. Covering problems are minimization problems and usually integer linear programs, whose dual problems are called packing problems. The most prominent
Covering_problems
Geometry problem on grid points
no-three-in-line problem and then scaling down the integer grid to fit within a unit square produces solutions to the Heilbronn triangle problem where the smallest
No-three-in-line_problem
Mathematical problem
for n {\displaystyle n} being any positive integer. The exercise of working through this problem may be used to explain and demonstrate exponents
Wheat_and_chessboard_problem
Even integers as sums of two primes
proved that every positive integer is the sum of four squares. See Waring's problem and the related Waring–Goldbach problem on sums of powers of primes
Goldbach's_conjecture
Mathematical problem of square numbers which are also square-pyramidal
Encyclopedia of Integer Sequences. OEIS Foundation. Weisstein, Eric W. "Square Pyramidal Number". MathWorld. Weisstein, Eric W. "Cannonball Problem". MathWorld
Cannonball_problem
Area of a right triangle with rational-numbered sides
x^{2}+ny^{2}=v^{2}} has a solution where x , y , u {\displaystyle x,y,u} , and v {\displaystyle v} are integers. Given a solution, the three numbers u 2
Congruent_number
Does the plane contains a dense set of points whose distances are all rational
its vertices, and then scaled to make the distances integers. However, like the Erdős–Ulam problem, Harborth's conjecture remains unproven. Anning, Norman
Erdős–Ulam_problem
River crossing puzzle
London: Routledge & Kegan Paul. pp. 4–5. Alcuin's Transportation Problems and Integer Programming Archived 2011-07-19 at the Wayback Machine, Ralf Borndörfer
Wolf, goat and cabbage problem
Wolf,_goat_and_cabbage_problem
Matrix form in linear algebra
matrices over the integers Z {\displaystyle \mathbb {Z} } . Just as reduced echelon form can be used to solve problems about the solution to the linear system
Hermite_normal_form
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001208 (solution to the postage stamp problem with 3 denominations
1000_(number)
Arrangement of points on a sphere
points goes to infinity, not for concrete values of N. The solution of the Thomson problem for two electrons is obtained when both electrons are as far
Thomson_problem
Type of Diophantine equation
nonsquare integer, and integer solutions are sought for x and y. In Cartesian coordinates, the equation is represented by a hyperbola; solutions occur wherever
Pell's_equation
Way to break a division problem into smaller steps
In arithmetic, short division is a division algorithm which breaks down a division problem into a series of easier steps. It is an abbreviated form of
Short_division
Mathematics of number-placement puzzle
related problem is the another solution problem (ASP): given an instance together with one solution, decide whether a second, distinct solution exists
Mathematics_of_sudoku
Natural number
not semiperfect. 70 is also part of the only nontrivial solution pair to the cannonball problem, along with 24. In Jewish tradition, Ptolemy II Philadelphus
70_(number)
On existence of a strongly regular graph
2014 as part of a set of problems posed in the DIMACS Conference on Challenges of Identifying Integer Sequences. Other problems in the set include the thrackle
Conway's_99-graph_problem
If G is a finitely generated group with exponent n, is G necessarily finite?
exponent, there exists a largest one. This provides a solution for the restricted Burnside problem for the case of prime exponent. (Later, in 1989, Efim
Burnside_problem
Problem in computer science
with a cardinality constraint. The maximum coverage problem can be formulated as the following integer linear program. The greedy algorithm for maximum coverage
Maximum_coverage_problem
identical-machines scheduling problem. The problem is parametrized by a positive integer k, and called k-way number partitioning. The input to the problem is a multiset
Multiway_number_partitioning
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SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
SHORT INTEGER-SOLUTION-PROBLEM
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