Seminar using Unsolved Problems in Number Theory
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Transcript of Seminar using Unsolved Problems in Number Theory
Seminar using Unsolved Problems in Number
Theory
Robert StyerVillanova University
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
Best Students
• Riemann Hypothesis and the connections with GUE theory in physics
• Birch & Swinnerton-Dyer Conjecture• Computing Small Galois Groups• Hilbert’s Twelfth Problem
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.
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”
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
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?
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) .
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.
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.
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. • Persistences in bases 2 through 12 are likely
1, 3, 3, 6, 5, 8, 8, 6, 7, 11, 13, 7.• Easy problem to understand and analyze; perfect
for an enthusiastic B-level major.
Gaussian Primes
• Student programmed very fast plotting of Gaussian primes
• Picture near origin• Red denotes central
member of a “Gaussian triangle”
• Analog of twin prime
Gaussian primes radius 10^5
•
Gaussian primes radius 10^15
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
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)
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!