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Further Generalizations Of Happy Numbers, E. Simonton Williams
Further Generalizations Of Happy Numbers, E. Simonton Williams
Rose-Hulman Undergraduate Mathematics Journal
A positive integer n is defined to be happy if iteration of the function taking the sum of the squares of the digits of n eventually reaches 1. In this paper we generalize the concept of happy numbers in several ways. First we confirm known results of Grundman and Teeple and establish further results extending the known structure of happy numbers to higher powers. Then we construct a similar function expanding the definition of happy numbers to negative integers. Working with this function, we prove a range of results paralleling those already proven for traditional and generalized happy numbers. Finally, …
Solution Of The Diophantine Equation (Maa+Nbb)=Cd(Mcc+Ndd) Using Rational Numbers, Georg Ehlers
Solution Of The Diophantine Equation (Maa+Nbb)=Cd(Mcc+Ndd) Using Rational Numbers, Georg Ehlers
Euleriana
This paper (E716) was published in Nova acta Academiae scientiarum imperialis petropolitanae, Volume 13 (1795/96), pp. 45-63. It was also included in Commentationes Arithmeticae, Volume II, as Number LXVIII, pp. 281-293 (E791). Euler starts with Fermat's Last Theorem and mentions the proofs for the cases n=3 and n=4 which he had completed himself earlier. He then moves on to make the sum of powers conjecture, which was later disproved in the second half of the 20th century. In this context he discusses his discovery of 134^4+133^4=158^4+59^4, which he calls unexpected. Euler derives the title equation from A^4+B^4=C^4+D^4, generalizing it to …
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Some Thoughts On The 3 × 3 Magic Square Of Squares Problem, Desmond Weisenberg
Rose-Hulman Undergraduate Mathematics Journal
A magic square is a square grid of numbers where each row, column, and long diagonal has the same sum (called the magic sum). An open problem popularized by Martin Gardner asks whether there exists a 3×3 magic square of distinct positive square numbers. In this paper, we expand on existing results about the prime factors of elements of such a square, and then provide a full list of the ways a prime factor could appear in one. We also suggest a separate possible computational approach based on the prime signature of the center entry of the square.
Euler Archive Spotlight, Erik R. Tou
Euler Archive Spotlight, Erik R. Tou
Euleriana
A survey of two translations posted to the Euler Archive in 2022.
Euler's Anticipations, Christopher Goff, Erik Tou
Euler's Anticipations, Christopher Goff, Erik Tou
Euleriana
Welcome to Volume 3 of Euleriana. This issue highlights occasions where Euler's work anticipated future results from other others, sometimes by decades or even centuries!
Unsolved Haiku, Scott W. Williams
Unsolved Haiku, Scott W. Williams
Journal of Humanistic Mathematics
This poem describes the still unsolved 1937 conjecture of Lloyd Collatz: Do repeated applications of the algorithm described yield the number 1?
The Genesis Of A Theorem, Osvaldo Marrero
The Genesis Of A Theorem, Osvaldo Marrero
Journal of Humanistic Mathematics
We present the story of a theorem's conception and birth. The tale begins with the circumstances in which the idea sprouted; then is the question's origin; next comes the preliminary investigation, which led to the conjecture and the proof; finally, we state the theorem. Our discussion is accessible to anyone who knows mathematical induction. Therefore, this material can be used for instruction in a variety of courses. In particular, this story may be used in undergraduate courses as an example of how mathematicians do research. As a bonus, the proof by induction is not of the simplest kind, because it …