Joy of Mathematical Experiments (Zoltán Galántai PhD)

"Like contemporary chemists—and before them the alchemists of old—who mix various substances together in a crucible and heat them to a high temperature to see what happens, today’s experimental mathematician puts a hopefully potent mix of numbers, formulas, and algorithms into a computer in the hope that something of interest emerges."
(Jonathan Borwein - Keith Devlin: The Computer as Crubicle)

New Observations on Lychrel Numbers and Palindromes (Aug 2021)
Details

The following empirical observations were made on the Lychrel-numbers: 1.Continuing the process that resulted in the first palindrome generated by the 196 algorithm starting from a certain number, probably we will find some new palindromes - but these palindrome sequences are only finite in length. 2.The palindromes generated by the 196 algorithm are clustered in bushes. In other words: certain sequences starting with different numbers end in the same palindrome. 3.There are more palindromes generated by the 196 algorithm with even than with odd ending.

Modified and Extended Ducci Sequences and their Inner Patterns (June 2020)
Details

A Ducci sequence can be interpreted as a subset of a broader category where not only the a < b < c < d order is allowed, but all of the possible order of the positive integers with different sizes. They are called “modified Ducci sequences”. Similarly, the notion of the “extended Ducci sequences” can be introduced adding one or more elements to the original, four-component Ducci sequence in ascending order (e.g. a < b < c < d < e). Hereby we are presenting conjectures about both of them.

Some inner structures of look-and-say sentences(Apr. 18. 2020)
Details

(Every look-and-say sequence has a typical, fractal-like structure if we write the first, second, third, etc. lines of a certain look-and-say sequence under each other.
All of the look-and-say sentences’ structures seem to be fundamentally similar.
Examining every look-and-say sentences up to one million; and then up to one billion randomly choosing one million numbers, roughly the 80 percent of every sentences’ first column (in short: FC) begins with the “n, 1, 1, 3” pattern where “n” is the first digit of the given number (e.g. 1 if the starting number is 1 or if the starting number is 119). 1,1, and 3 are the starting digits of the second, third and fourth rows. Here “n” is called “head” and the infinitely repeating pattern is the “tail”.
Sooner or later the FC of every look-and-say sequence takes the 1,1,3 tail and, from that point, this pattern is repeating to the infinity.
In the case of a second-generation FC (where the inputs, instead of the look-and-say sequences, the results of the comparisons of two look-and-say sequences), the tail is either 0,0,0 or 0,2,2.
It is plausible that every tail is 3 digit long and there are only three tail versions.
Comparing the digits of two look-and-say sequences in their overlapping parts, two main types can be identified: Where the number of the non-equal digits (difference, or, in short, DIFF) of the same rows in the same position is almost zero; and where the digits of the two look-and-say sequences mainly different.
A DIFF converges to a certain value if the tail of the comparison of two look-and-say sequences is 0,0,0. Otherwise, it converges to infinity.
The DIFFs of some different pairs are identical, thus either there are some pairs where this equality exists, or every pairs’ DIFFs belongs to a group where the DIFFs are the same.

On the Lenghts of the Look-and-say sequences (Jan. 02. 2020)
Details

1. The growth of the lengths of the new lines of a sequence can be described by an exponential formula. Since the lengths of the sequences with different starting values produce non-totally identical growths, we have to use slightly different formulas.
2. The lengths (number of the digits) in the nth line of the sequences do not follow a simple function: their pattern is fractal-like

Shifted Narayana Sequences: A Short Note (Sept. 09. 2019)
Details

The logic of the shifted Narayana's cow sentences is similar to the other Fiboiacci-like ones. We are summarizings some numerical results.

SHIFTED FIBONACCI-, TRIBONACCI-, N-NACCI SENTENCES; THEIR CONSTANTS AND THE RULES REGULATE THEM (Sept. 09. 2019)
This is a summary about the shifted Fibonacci-sequences and their derivatives (Tribonacci,Tetranacci… etc. sequences) to show the existence of the constants of the shifted n-nacci sequencesand to give a short description about those mathematical rules that determine the values of theseconstants. In sum, the shifted n-nacci sequences are governed by the same mathematical regularitiesas the Fibonacci sequence.
Details

On the Fine Structure of Shifted Padovan, Perrin, and Fibonacci Sequences and Lucas Series (Aug. 23. 2019)
The aim of this short paper to introduce the notion of the shifted sequences; and to describe the mathematical rules determine the constants of the shifted sequences of the Padovan, Perrin and Fibonacci sequences and Lucas series. These new constants (similar to to plastic number) are listed.
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Observation about Fibonacci Sequences and Lucas Series: Some New Constants (Aug. 12. 2019)
This short paper is to present some connections between the Fibonacci numbers; and between the Fibonacci sequences and the Lucas series.It is not our aim to exhaust this subject exploring every possible consequence thus we present only some examples. In parallel, some new constants will be introduced.
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Possible interpretations of Turing's halting problem (Sept. 08. 2018)
Turing originally examined the halting problem to find an answer for Hilbert’s Entscheidungsproblem and his conclusion was roughly that since it is impossible to decide whether a program would halt or run forever, there is no solution for Hilbert’s problem. The typical argumentation for this undecidability is based on a contradiction of a self-referential logical paradox. Our aim is to point out that contrary to the widely accepted opinion, even if we accept the existence of the paradox, it can be stated that thanks to the new results of philosophy of the science that has appeared since the publication of Turing’s work there is more than one possible interpretation for the halting problem and there is more than one possible implementation if we want to write a program. Thus it is not impossible to decide whether an arbitrary program would halt within finitely many steps if we accept certain starting points.
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Divisibility as graph (Aug. 19. 2018)
We can classify all possible graphs of the numbers of divisors into four categories.
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Generalization of the Goldbach conjecture (May. 19. 2018)
We can generalize the Goldbach conjecture for the basic mathematical operations.
Details

A numerical correlation between the numbers of divisors; the value of the muptiplication of the divisors, and the value of the number (Apr. 22. 2018)
In short:It is true for every positive integer (except for those numbers where the number of the divisors is odd and where another rule applies) that N=n^(d(n)/2).
Details

An alternative to the Goldbach conjecture (Apr. 4. 2018)
In short: not only the addition of the primes is a saving solution: (prime*prime)+prime also meets the purpose.
Details

Divisibility of the amicable pairs, sociable chains, and the sums of the natural numbers and their proper divisors by ten (Nov. 11. 2017)
In short: the larger the number, the more likely that the sum of the number and its proper divisors can be divided by ten. Similar rules are observed for amicable numbers and sociable chains,
Details and OEIS

Conjecture of the sum of sociable number cycles (Sept. 5. 2017)
"As the number of sociable number cycles with length greater than 2 approaches infinity, the percentage of the sums of the sociable number cycles divisible by 10 approaches 100%"
OEIS

Sum of amicable pairs conjecture (Sept. 5. 2017)
"As the number of the amicable numbers approaches infinity, the percentage of the sums of the amicable pairs divisible by 10 approaches 100%."
Sum of amicable pairs conjecture (.pdf)
OEIS

Mining patterns of the digital numbers to solve equations (June 17. 2017)
Constructing numbers using simple patterns of zeros and ones in binary system can make unnecessary some more or less complicated equations. Hereby I am presenting some findings as a result of a few hour search. The easiness of finding these binary patterns leads to two questions. First of all, whether it means that it is really easy to find solutions for certain types of equations. Another question is whether there are opportunities to solve certain equations in other number systems (e.g. in ternary or hexadecimal) using this approach. I focus only for the binary system now, but, obviously, a more systematic research is needed.
(.pdf)

Sum of Divisors Conjecture (May 29. 2017)
Introducing a new conjecture stating that every positive integer larger than 8 can be constructed as a sum of the sums of two other numbers’ proper divisors. This conjecture is similar to Goldbach’s conjectures.
Full paper
(.pdf)

Problems with Cantor’s Infinities (March 21. 2017)
According to Cantor, it can be shown by the diagonal method that there are infinitely more real than natural numbers since it is impossible to create a one-to-one correspondence (bijection) between them, and there are always real numbers without a natural number assigned to them. Obviously, the definition of the real and the definition of the natural numbers have a fundamental role in this case, as the definition prescribes their features–so new definitions of the natural numbers would change their nature. It is important to emphasize this, since the nature of natural numbers determined by the definition has fundamental role in the size of both the sets and the power sets. Cantor stated that the size of the latter was infinitely larger, but if we accept the definition of the natural numbers in its actual form, then Cantor’s statement cannot be true. Besides examining this problem, we shall point out that a modified definition of the natural numbers makes possible a one-to-one correspondence between natural and real numbers, but unable to cause difference in the size of the sets and power sets.
Full paper
(.pdf)

On Twin Amicable Numbers (May 18. 2016)
The aim of this short paper is to examine twin amicable numbers using a modified analogy of the twin primes and to present some numerical observations both on the distribution of twin amicable numbers and on the degree of relationship of the amicable numbers. This article contains both the list of known twin amicable numbers and the list of double, triple etc. twin amicable numbers.
On Twin Amicable Numbers (.pdf)
software (source code in Clipper by Béla Galántai)
raw data (.xlsx)
OEIS

This work by Zoltán Galántai PhD is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.