View Proposal #548
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| ID | 548 |
|---|---|
| First Name | Christian |
| Last Name | Roettger |
| Institution | Iowa State University |
| Speaker Category | faculty |
| Title of Talk | Balanced Numbers and Balanced Primes |
| Abstract | Balanced numbers are odd natural numbers n which have an equal number of 0s and 1s in the periodic part of the base-2 representation of their reciprocal 1/n. We present some insights about balanced numbers that use just elementary Number Theory like the Quadratic Reciprocity Theorem. In particular, if a prime p is congruent to 3 or 5 modulo 8, then p is balanced. If a prime p is congruent to 7 modulo 8, then p is not balanced. All powers of p are balanced iff p is. The case of primes congruent to 1 modulo 8 is much more difficult. Hasse made a breakthrough in 1966, showing that the balanced primes have a Dirichlet density of 17/24. We have refined Hasse's result slightly. Another question is how big is the set of balanced numbers (not only primes) less than x? Using a method due to Landau, we can show that this is bounded above by C x/log^(1/4) (x) and below by D x / log^(3/4)(x), with constants C, D > 0. I solemnly promise that I won't go into the gory detail, only highlight the beautiful and accessible parts of the subject. The second part of the talk is joint work with Joshua Zelinsky. |
| Subject area(s) | |
| Suitable for undergraduates? | ? |
| Day Preference | SaturdayMild |
| Computer Needed? | |
| Bringing a laptop? | |
| Overhead Needed? | |
| Software requests | |
| Special Needs | |
| Date Submitted | 2019-10-13 11:23:57 |
| Year | 2019 |