Related papers: The birthday boy problem
In this paper we find an identity that gives a representation for the logarithm of any two irrational numbers $a, b >1$ in terms of a series whose terms are ratios of elements from the Beatty Sequences generated by these two numbers. We…
We investigate a variation of Nicomachus's identity in which one term in the cubic sum is replaced by a different cube. Specifically, we study the Diophantine identity \[ \sum_{j=1}^{n} j^3 + x^3 - k^3 = \left( \sum_{j=1}^{n} j + x - k…
We present a new fast algorithm for computing the Boys function using a nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are…
This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the…
The continued fraction expansion of an irrational number $\alpha$ is eventually periodic if and only if $\alpha$ is a quadratic irrationality. However, very little is known regarding the size of the partial quotients of algebraic real…
We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the…
For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd…
Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by…
The current work revisits the results of L.F. Meyers and R. See in [3], and presents the census-taker problem as a motivation to introduce the beautiful theory of numbers.
We prove the second author's "denominator conjecture" [40] concerning the common denominators of coefficients of certain linear forms in zeta values. These forms were recently constructed to obtain lower bounds for the dimension of the…
A basic question of Diophantine approximation, which is the first issue we discuss, is to investigate the rational approximations to a single real number. Next, we consider the algebraic or polynomial approximations to a single complex…
These lecture notes cover classical undecidability results in number theory, Hilbert's 10th problem and recent developments around it, also for rings other than the integers. It also contains a sketch of the authors result that the integers…
We provide both human and computer (even better collaboration between the two) proofs to four recent American Mathematical Monthly problems, namely problem 11897, problem 11899, problem 11916, and problem 11928. We also show that problem…
We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A…
D. Wallace has tried to use decoherence to solve the preferred basis problem of Everettian Quantum Mechanics, and this solution lays the foundation for his proof of the Born rule. But this is a circular argument, as approximations used in…
V.I. Arnold has experimentally established that the limit of the statistics of incomplete quotients of partial continued fractions of quadratic irrationalities coincides with the Gauss--Kuz'min statistics. Below we briefly prove this fact…
This article, based on a talk, treats some elementary, but not completely simple examples from probability. They concern multiple birthday coincidences, throwing dice, the combinatorics of the German card game "Doppelkopf", and the…
In this paper, we give a brief survey of recent developments on Noether's problem and rationality problem for multiplicative invariant fields including author's recent papers Hoshi [Hos15] about Noether's problem over Q, Hoshi, Kang and…
Ratio statistics--such as relative risk and odds ratios--play a central role in hypothesis testing, model evaluation, and decision-making across many areas of machine learning, including causal inference and fairness analysis. However,…
We announce a number of conjectures associated with and arising from a study of primes and irrationals in $\mathbb{R}$. All are supported by numerical verification to the extent possible.