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
• 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
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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
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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!