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Seminar using Unsolved Problems in Number Theory

Seminar using Unsolved Problems in Number Theory. Robert Styer Villanova University. Seminar. Textbook: Richard Guy’s Unsolved Problems in Number Theory (UPINT) About 170 problems with references

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Seminar using Unsolved Problems in Number Theory

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  1. Seminar using Unsolved Problems in Number Theory Robert Styer Villanova University

  2. Seminar • Textbook: Richard Guy’s Unsolved Problems in Number Theory (UPINT) • About 170 problems with references • Goals of seminar: Experience research Use MathSciNet and other library tools Experience giving presentations Writing Intensive: must have theorem/proof

  3. Best Students • Riemann Hypothesis and the connections with GUE theory in physics • Birch & Swinnerton-Dyer Conjecture • Computing Small Galois Groups • Hilbert’s Twelfth Problem

  4. Regular Math Majors • Happy Numbers • Lucky Numbers • Ruth-Aaron numbers • Persistence of a number • Mousetrap • Congruent numbers • Cute and obscure is good! Room to explore.

  5. What do these students accomplish? • Happy Numbers, UPINT E34 • 44492 -> 4^2 + 4^2 + 4^2 + 9^2 + 2^2 =133 -> 1^2 + 3^2 + 3^2 =19 -> 1^2 + 9^2 = 82 -> 70 -> 49 -> 97 -> 130 -> 10 -> 1 -> 1 -> 1 … • Fixed point 1, so 44492 is “happy”

  6. Happy Numbers • 44493 -> 4^2 + 4^2 + 4^2 + 9^2 + 3^2 = 138 -> 74 -> 65 -> 61 -> 37 -> 58 -> 89 -> 145 -> 42 -> 20 -> 4 -> 16 -> 37 -> 58 -> … • A cycle of length 8, so 44493 is “unhappy” (or “4-lorn”) • Most numbers (perhaps 6 out of 7) seem to be unhappy

  7. Happy Numbers • Obvious questions: Any other cycles? (no)Density of happy numbers? (roughly 1/7?)What about other bases? Consecutive happy numbers? • 44488, 44489, 44490, 44491, 44492 first string of five consecutive happy numbers. • What is the first string of six happy numbers?

  8. Happy Numbers • A proof that there are arbitrarily long strings published by El-Sedy and Siksek 2000. • A student inspired me to find the smallest example of consecutive strings of 6, 7, 8, 9, 10, 11, 12, 13 happy numbers. • This year a student found the smallest examples of 14 and 15 consecutive happy #s. • Order the digits (note 16 -> 37 also 61 -> 37) .

  9. 14 Consecutive Happy Numbers • My old method for 14 would need to check about 10^15 values • Ordering the digits made his search 7 million times more efficient. • Students enjoy doing computations • There are always computational questions that no one has bothered doing, and they are perfect for students.

  10. Multiplicative Persistence • Another digit iteration problem: multiply the digits of a number until one reaches a single digit. UPINT F25 • 6788 -> 6*7*8*8 = 2688 -> 2*6*8*8 = 768 -> 7*6*8 = 336 -> 3*6*6 = 108 -> 0. • 6788 has persistence 5 • Maximum persistence? • Sloane 1973 conjectured 11 is the maximum.

  11. Multiplicative Persistence • Sloane calculated to 10^50 • My student calculated much higher and also for other bases. • Conjecture holds up to 10^1000 in base 10, and similar good bounds for bases up to 12. • Persistencesin bases 2 through 12 are likely1, 3, 3, 6, 5, 8, 8, 6, 7, 11, 13, 7. • Easy problem to understand and analyze; perfect for an enthusiastic B-level major.

  12. Gaussian Primes • Student programmed very fast plotting of Gaussian primes • Picture near origin • Red denotes centralmember of a “Gaussian triangle” • Analog of twin prime

  13. Gaussian primes radius 10^5

  14. Gaussian primes radius 10^15

  15. Questions about Gaussian primes • Density, analog of the density of primes • Density of triangles, analog of the density of twin primes • “Moats:” the student estimated what radius should allow a larger moat than those proven in the literature, and he drew pictures showing typical densities at that radius

  16. Other simple problems • Epstein’s Put or Take a Square Game: new bounds, replaced “square” with “prime,” “2^n” • Euler’s Perfect Cuboid problem: use other geometric figures, what subsets of lengths can one make rational • Twin primes: other gaps between primes • N queens problem: use other pieces • Egyptian fractions: conjectures on 4/n and 5/n, what about higher values like 11/n? • Practically perfect numbers |s(n)-2n| < sqrt(n)

  17. Summary • Simple problems work well • Obscure problems have more room to explore • Students can compute new results if one looks for specific instances of general theory: least example of n consecutive happy numbers persistence in several bases density of Gaussian prime triangles • Students love finding something that is their addition to knowledge!

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