#A3 INTEGERS 12A (2012): John Selfridge Memorial Issue PERFECT POWERS WITH FEW TERNARY DIGITS
نویسندگان
چکیده
We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically. –Dedicated to the memory of John Selfridge.
منابع مشابه
#A16 INTEGERS 12A (2012): John Selfridge Memorial Issue THE SEARCH FOR AURIFEUILLIAN-LIKE FACTORIZATIONS
We searched the Cunningham tables for new algebraic factorizations similar to those discovered by Aurifeuille. A naive search would have been too slow. We accelerated it enough to make it feasible. Many interesting results were found. –Dedicated to the memory of John Selfridge, who loved the integers.
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We classify all integer squares (and, more generally, q-th powers for certain values of q) whose ternary expansions contain at most three digits. Our results follow from Padé approximants to the binomial function, considered 3-adically.
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We show how to obtain the solutions of families of systems of two Pell equations; these families are parameterized by the prime numbers. -Dedicated to the memory of John Selfridge. His opera voice is no more, but his voice in mathematics will continue to be heard, powerful.
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We consider the second of Mullin’s sequences of prime numbers related to Euclid’s proof that there are infinitely many primes. We show in particular that it omits infinitely many primes, confirming a conjecture of Cox and van der Poorten.
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An idea used in the characterization of even perfect numbers is used, first, to derive new necessary conditions for the existence of an odd perfect number and, second, to show that there are no even 3-perfect numbers of the form 2aM , where M is odd and squarefree and a ≤ 718, besides the six known examples. –In memory of John Selfridge
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