Abstract
In this paper, we discuss strings of 3’s and 7’s, hereby dubbed “dreibens.” As a first step towards determining whether the set of prime dreibens is infinite, we examine the properties of dreibens when divided by 7. by determining the divisibility of a dreiben by 7, we can rule out some composite dreibens in the search for prime dreibens. We are concerned with the number of dreibens of length n that leave a remainder i when divided by 7. By using number theory, linear algebra, and abstract algebra, we arrive at a formula that tells us how many dreibens of length n are divisible by 7. We also find a way to determine the number of dreibens of length n that leave a remainder i when divided by 7. Further investigation from a combinatorial perspective provides additional insight into the properties of dreibens when divided by 7. Overall, this paper helps characterize dreibens, opens up more paths of inquiry into the nature of dreibens, and rules out some composite dreibens from a prime dreiben search.
Highlights
Introduction to dreibens and the search for infinitely many prime dreibensThe existence of infinitely many prime numbers is an elementary fact of number theory
We discuss strings of 3’s and 7’s, hereby dubbed “dreibens.” As a first step towards determining whether the set of prime dreibens is infinite, we examine the properties of dreibens when divided by 7. by determining the divisibility of a dreiben by 7, we can rule out some composite dreibens in the search for prime dreibens
Linear algebra, and abstract algebra, we arrive at a formula that tells us how many dreibens of length n are divisible by 7
Summary
The existence of infinitely many prime numbers is an elementary fact of number theory. Let us define Aij(n) as the number 1 of dreibens of length n that leave a remainder i when divided by j. Let us consider a dreiben Dn+1 of length n + 1, which could leave a remainder of 0, 1, 2, 3, 4, 5, or 6 when divided by 7. "We can find formulas for Ai7(n + 1) by considering the first n digits of a dreiben of length n + 1, seeing what the last digit must be such that the dreiben leaves a remainder i when divided by 7." 91. We get the following formulas for Ai7(n + This linear system allows us to quickly find the number of dreibens of length n that leave a remainder i when divided by 7. A is a diagonalizable matrix, so An is diagonalizable: no matter what n is, the first row of the matrix An is always
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