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!