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[7] Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics". [8] However, though the Collatz conjecture itself remains open, efforts to solve the problem have led to new techniques and many partial results. [8] [9]
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.
Lothar Collatz (German:; July 6, 1910 – September 26, 1990) was a German mathematician, born in Arnsberg, Westphalia. The "3x + 1" problem is also known as the Collatz conjecture, named after him and still unsolved. The Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a positive square matrix was also named after him.
Once known as the "Weil conjecture". 2001: Mark Haiman: n! conjecture: representation theory: 2001: Daniel Frohardt and Kay Magaard [17] Guralnick–Thompson conjecture: monodromy groups: 2002: Preda Mihăilescu: Catalan's conjecture, 1844: exponential diophantine equations: ⇐Pillai's conjecture⇐abc conjecture Mihăilescu's theorem 2002
The Erdős–Turán conjecture on additive bases of natural numbers. A conjecture on quickly growing integer sequences with rational reciprocal series. A conjecture with Norman Oler [2] on circle packing in an equilateral triangle with a number of circles one less than a triangular number. The minimum overlap problem to estimate the limit of M(n).
The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r , then the L -function L ( E , s ) associated with it vanishes to order r at s = 1 .
Juggler sequences therefore present a problem that is similar to the Collatz conjecture, about which Paul Erdős stated that "mathematics is not yet ready for such problems". For a given initial term n , one defines l ( n ) to be the number of steps which the juggler sequence starting at n takes to first reach 1, and h ( n ) to be the maximum ...
For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 10 12 (1.2 trillion). However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but ...