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Landau's fourth problem asked whether there are infinitely many primes which are of the form = + for integer n. (The list of known primes of this form is A002496 .) The existence of infinitely many such primes would follow as a consequence of other number-theoretic conjectures such as the Bunyakovsky conjecture and Bateman–Horn conjecture .
The Pell solution (19,6) leads to the pair of consecutive P-smooth numbers ; the other two solutions to the Pell equation do not lead to P-smooth pairs. For q = 6, the first three solutions to the Pell equation x 2 − 12y 2 = 1 are (7,2), (97,28), and (1351,390). The Pell solution (7,2) leads to the pair of consecutive P-smooth numbers .
The history of the problem dates back at least to Gersonides, who proved a special case of the conjecture in 1343 where (x, y) was restricted to be (2, 3) or (3, 2).The first significant progress after Catalan made his conjecture came in 1850 when Victor-Amédée Lebesgue dealt with the case b = 2.
Since this number is large for large , this lends credence to Legendre's conjecture. [4] It is known that the prime number theorem gives an accurate count of the primes within short intervals, either unconditionally [ 5 ] or based on the Riemann hypothesis , [ 6 ] but the lengths of the intervals for which this has been proven are longer than ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.In his 1972 paper, "Reducibility Among Combinatorial Problems", [1] Richard Karp used Stephen Cook's 1971 theorem that the boolean satisfiability problem is NP-complete [2] (also called the Cook-Levin theorem) to show that there is a polynomial time many-one reduction ...
Complete coloring, a.k.a. achromatic number [3]: GT5 Cycle rank; Degree-constrained spanning tree [3]: ND1 Domatic number [3]: GT3 Dominating set, a.k.a. domination number [3]: GT2 NP-complete special cases include the edge dominating set problem, i.e., the dominating set problem in
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form (+). [1] The study of these numbers dates back to Aristotle.They are also called oblong numbers, heteromecic numbers, [2] or rectangular numbers; [3] however, the term "rectangular number" has also been applied to the composite numbers.