- Our research focused on the so-called Diophantine equations. These are equations with multiple unknowns, for which the solutions are sought in the set of integers. One of the best-known examples would be the equation a2 + b2 = c2, which is taught in high school, where, for example, the three numbers (3, 4 and 5) give a correct solution (32 + 42 = 52), but this is only one of an infinite number of solutions- said István Pink, who had, in fact, worked on a a much more general problem together with his Japanese colleague. They explored the equation ax+by=cz, where there are unknowns even in the coefficients.
Their question was the following: if we take three arbitrarily fixed relative primes greater than one – i.e., positive integers a, b, and c that have no common divisor – and raise them to arbitrary positive integer powers x, y, and z, how many times would the sum of two of them exactly equal the third?
The associate professor of the Institute of Mathematics at the Faculty of Sciences and Technology, UD, and his co-author proved that, no matter what numbers a, b, and c we pick for the basis, the equation ax+by=cz can only have two (x,y,z) solutions at most among positive integers, except for the cases (a,b,c)=(3,5,2) and (5,3,2), when the equation has exactly three solutions. This proved to be their correct answer to a question that has been open for decades.
- The real value of this result is that it is as accurate as possible in the sense that I can choose the integers a,b,c in an infinite number of ways so that the equation should have exactly two (x,y,z) solutions- said the UD’s researcher.
Their success did not come about by chance. István Pink is a member of the highly renowned Debrecen School of Number Theory, associated with the name of Professor Kálmán Győry. The researcher from Debrecen and his colleague worked on the solution for two years, exchanging more than a thousand emails during that time. There was a point in time when they were almost satisfied with a weaker statement, but their perseverance and belief in mathematics helped them to move over that standstill.
- I did feel the power: I also felt that this had to be done- recalled the researcher. In the end, the proof became 78 pages long and was reviewed for two years before publication, instead of the usual couple of months. The D1-rated study ultimately gained serious international recognition, as it was published in one of the most prestigious journals on math, the American Journal of Mathematics. Not surprisingly, it also won the Count István Tisza Foundation for the University of Debrecen Publication Award.
Although this is considered basic research, number theory is indeed the foundation of all modern digital security. When we pay with a credit card or send an encrypted message, similarly complex number theory algorithms protect our data in the background. István Pink's findings may not directly contribute to the development of tomorrow’s smartphones, yet he has provided science with new methods that may become part of fundamental technologies in fifty or a hundred years.
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