Related papers: The Third Trick

In this paper, we prove certain theorems about three consecutive primes.

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

We prove some new results related to Tanaka's formula.

We prove several congruences for trinomial coefficients.

We prove Simon's conjecture for 3-manifolds.

We compute the derived functors of the third symmetric-power functor and their cross-effects for certain values. These calculations match predictions by the first named author and largely prove them in general.

We establish an equivalent condition to the validity of the Collatz conjecture, using elementary methods. We derive some conclusions and show several examples of our results. We also offer a variety of exercises, problems and conjectures.

We define a function by refining Stern's diatomic sequence. We name it the {\it assembly function}. It is strictly increasing continuous. The first and the second main theorems are on an action to the function. The third theorem is on…

We investigate a result on convergence of double sequences of numbers and how it extends to measurable functions.

New expansions of the number zeta(3) in continuous fractions are found.

We develop a theory of modulus triples, for future motivic applications.

We define and prove isomorphisms between three combinatorial classes involving labeled trees. We also give an alternative proof by means of generating functions.

We prove a discrete approximation of functionals with jumps and creases.

Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence…

In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of…

Some new identities for Schur functions are proved. In particular, we settle in the affirmative a recent conjecture of Ishikawa-Wakayama and solve a problem raised by Bressoud.

We provide complete proofs of the lemmas about the properties of the regularized loss function that is used in the second order techniques for learning time-series with structural breaks in Osogami (2021). In addition, we show experimental…

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

Taking a new approach towards analyzing the Collatz Problem, or, 3x+1 conjecture. Introducing some new functions, the Collatz-2 and Collatz-3 sequences, as well as deducing results related to Collatz-2 and Collatz-3 sequences.

We propose a regularization technique and apply it to the Euler product of zeta functions, mainly of the Riemann zeta function, to make unknown some clear. In this paper that is the first part of the trilogy, we try to demonstrate the…