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)


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

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