# Seminar using

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Seminar using Unsolved Problems in Number Theory

Robert Styer

Villanova University

Slide2Seminar

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

Slide3Best Students

Riemann Hypothesis and the connections with GUE theory in physics

Birch &

Swinnerton

-Dyer Conjecture

Computing Small Galois Groups

Hilbert’s Twelfth Problem

Slide4Regular Math Majors

Happy Numbers

Lucky Numbers

Ruth-Aaron numbers

Persistence of a number

Mousetrap

Congruent numbers

Cute and obscure is good! Room to explore.

Slide5What 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”

Slide6Happy 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

Slide7Happy Numbers

Obvious questions:

Any other cycles? (no)

Density of happy numbers? (roughly 1/7?)

What about other bases?

C

onsecutive happy numbers?

44488, 44489, 44490, 44491, 44492 first string of five consecutive happy numbers.

What is the first string of six happy numbers?

Slide8Happy 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.

O

rder

the digits

(note 16

-> 37 also 61 -> 37) .

Slide914 Consecutive Happy Numbers

My old method

for 14 would need

to check about 10^15

values

O

rdering 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

.

Slide10Multiplicative 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.

Slide11Multiplicative 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.

Slide12Gaussian Primes

Student programmed very fast plotting of Gaussian primesPicture near originRed denotes centralmember of a “Gaussian triangle”Analog of twin prime

Slide13Gaussian primes radius 10^5

Slide14

Gaussian primes radius 10^15

Slide15Questions about Gaussian primes

D

ensity, 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

Slide16Other 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)

Slide17Summary

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!

## Seminar using

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