Catalan ’ s Conjecture
نویسنده
چکیده
منابع مشابه
The First Function and Its Iterates
Let s(n) denote the sum of the positive divisors of n except for n iteself. Discussed since Pythagoras, s may be the first function of mathematics. Pythagoras also suggested iterating s, so perhaps considering the first dynamical system. The historical legacy has left us with some colorful and attractive problems, mostly still unsolved. Yet the efforts have been productive in the development of...
متن کاملThe Aliquot Constant, after Bosma and Kane
Let s(n) be the sum of those positive divisors of the natural number n other than n itself. A conjecture of Catalan–Dickson is that the “aliquot” sequence of iterating s starting at any n terminates at 0 or enters a cycle. There is a “counter” conjecture of Guy–Selfridge that while Catalan–Dickson may be correct for most odd numbers n, for most even seeds, the aliquot sequence is unbounded. Len...
متن کاملRELAXATIONS OF THE ABC CONJECTURE USING INTEGER k ’TH ROOTS
Weakened forms of the ABC conjecture are defined in terms of the upper k’th root functions. These weakened forms, with quite small explicit values of their parameters, are shown to imply the asymptotic Fermat, Beale, general Fermat, and Catalan conjectures, that there exist an infinite number of non–Wieferich primes, that there exist only finitely many consecutive powerful numbers, Hall’s conje...
متن کاملSQUARE q , t - LATTICE PATHS AND ∇ ( p n ) NICHOLAS A
The combinatorial q, t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q, t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n’th q, t-Catalan number is the Hilb...
متن کاملOn the H-triangle of generalised nonnesting partitions
With a crystallographic root system Φ, there are associated two Catalan objects, the set of nonnesting partitions NN(Φ), and the cluster complex ∆(Φ). These possess a number of enumerative coincidences, many of which are captured in a surprising identity, first conjectured by Chapoton. We prove this conjecture, and indicate its generalisation for the Fuß-Catalan objects NN (Φ) and ∆(Φ), conject...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009