## Mathematician's Secret Room

### Contents

What's new (July 22, 2009)

Chapter 0 : Opening --- Why I am interested in Number Theory.

Chapter 1 : 4/n = 1/a + 1/b + 1/c [D11]
Chapter 2 : Squares consisted of 3 different digits [F24]
Chapter 3 : n = (x + y + z)(1/x + 1/y + 1/z)
Chapter 4 : n = x3 + y3 + z3 [D5]
Chapter 5 : Repeating Decimals [A3]
Chapter 6 : Additive Palindromicness of Natural Numbers [F32]
Chapter 7 : Collatz's Conjecture [E16]
Chapter 8 : Continued Fraction and Pell's Equation
Chapter 9 : Amicable Numbers [B4][B5]
Chapter 10 : Congruent Numbers (Congruum) [D27]
Chapter 11 : Number Theoretic Algorithms [A3] [B45] [F10]
Chapter 12 : Integer Factorization Algorithms

Appendix 1 : WIFC (World Integer Factorization Center)
Appendix 2 : Bibliography
Appendix 3 : How to join in the Factorization Project
Appendix 4 : Benchmark of Tomabechi's "ppmpqs.exe"
Appendix 5 : (new !) Prime table (Primes near to 2n, 10n, factorial, primorial and compositorial)

[nn] denote the number in Richard K. Guy, "Unsolved Problems in Number Theory" Third Edition.

### Abstracts

#### Chapter 1 : 4/n = 1/a + 1/b + 1/c (D11 Egyptian fractions)

1. (A few contents are still written in Japanese)

2. Erdös and Strauss conjectured that the following Diopantine equation

4/n = 1/a + 1/b + 1/c
could be solved in positive integers for all n < 1.

3. I found how to construct the parameterize solution from arbitrary solutions.

 Theorem : Let A, B, C in N be a solution of the Diophantine equation m/P=1/A+1/B+1/C, B=kP (A
If only C is divisable by P, then there is another way to construct the parameterize solution heuristically.
The program is here.
This will help to extend the search range.

4. All solution of m/P=1/A+1/B+1/C, m=4,5,6,7, P=prime less than 100
and derived parameterize solutions are shown at,
4/P=1/A+1/B+1/C : P=2..47, P=53..97
5/P=1/A+1/B+1/C : P=2..47, P=53..97
6/P=1/A+1/B+1/C : P=2..47, P=53..97
7/P=1/A+1/B+1/C : P=2..47, P=53..97

#### Chapter 2 : Squares consisted of 3 different digits (F24 Some decimal digital problems)

1. Japanese mathematician Sin Hitotsumatsu asked the proof or the contradiction that, apart from tribial solutions 102n, 4*102n and 9*102n, there are perfect square number consisted of only 2 different decimal digits. Known largest solution is

816192 = 6661661161.

2. I extended the condition of this problem as (i) 3 different digits (ii) perfect n-th power, and found many remarkable parametrized patterns and sporadic solutions.
All solutions of sporadic pattern up to 1025 for non-zero patterns
and up to 1023 patterns including zero are here.

1501674067666649999852= 22550250055025025225250200022000050000225 (April 07, 2008)
1490670655108730886732= 22220990020022929092929022220290920900929 (April 07, 2008)

31802522547770395385022= 10114004404014444004140001011411401140404004 (April 28, 2008)
66749834797132300059622= 44555404454444540454555540045000554555545444 (June 02, 2008)
30157752651590112301382= 9094900449944904494440090444449999999499044 (June 09, 2008)

449499949999999499999952= 2020502050500020505000050500052500000500000025 (October 31, 2008)
208327397238179751383622= 434003044400343443044430000434430333044043044 (October 31, 2008)
All solutions of infinite pattern (parameterized solutions) are here.

3. Sometimes the first (smallest) solution becomes very large. For example,

056 : 22360814084166662 = 5000060065066660656065066555556 (May 4, 1997)
079 : 88191722853734972 = 77777799799099990007000790009009 (May 5, 1997)
789 : 99493707779879172 = 98989978877879888789778997998889 (May 10, 1997)
019 : 436942788245669642512 = 1909190001999001011109190090109911991001 (May 6, 1998)

4. I searched up to the following range, but couldn't find the solution of 013 and 689;

013 : 1024
678 : 1025

5. The solutions for higher powers are here.
I couldn't find the perfect 7-th power consisted of 3 different digits.

#### Chapter 3 : n=(x+y+z)(1/x+1/y+1/z)

1. Searching for integer solutions of above equation for abs(n) is less than 100 by primitive way.
Solutions -100 ≤ n ≤ 100 (by exhaustive search) are here.

2. Solutions with Cremona's mwrank

Solutions -500 ≤ n ≤ -1
Solutions 1 ≤ n ≤ 500
Now you can get larger solutions like
n = -100 : (x,y,z) = (4450012553, 219887106322, -663397965750)
n = 94 : (x,y,z) = (571064, -1799160, 79045681)

3. Searching for integer solutions of the equation x/y+y/z+z/x=n
for abs(n) is less than 100 by primitive way.
Solutions -100 ≤ n ≤ 100 (by exhaustive search) are here.

4. Solutions with Cremona's mwrank

Solutions for -100 ≤ n ≤ -1
Solutions for 1 ≤ n ≤ 100
Now you can get larger solutions like

n = -48 : (x,y,z) = (72072752816411426700, 33132848506525529596688, -2507202774146263930905)
n = 62 : (x,y,z) = (4467832378776170000, -51609086900999886977, 278221158496143039700)

#### Chapter 4 : n = x3 + y3 + z3 (D5 Sum of four cubes)

1. Searching for above equation's integer solutions of n less than 10000.

2. Solutions are here.
Search range is now max { |x| , |y| , |z| } ≤ 106.
For certain numbers, search range is up to 1010 and beyond,
which achieved by Dan J. Bernstein and Noam D. Elkies.

3. And solutions of n = x3 + y3 + 2z3 are here.
Search range is now max { |x| , |y| , |z| } ≤ 106.
Jean-Charles Meyrignac and Mike Oakes tried up to max { |x| , |y| , |z| } ≤ 4*106.

#### Chapter 5 : Repeating Decimals (A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k.2n+1)

1. Introduction of Repeating Decimals (this is not an unsolved problem).

#### Chapter 6 : Additive Palindromicness of Natural Numbers (F32 Conway's RATS and palindromes)

1. (A few contents are still written in Japanese)

2. Almost all nutural number seems to become palindromic after several iteration of addition with reverse order of itself.

3. 196 is not known whether it will become palindromic or not.

4. And there are some numbers which will become palindromic after several iteration of addition but it takes many times. For example, 89 requires 24 times iterations.
The largest known iteration times is 10000000525586206 (232 times) found by Helmut Postl.
Other examples are listed here.

#### Chapter 7 : Collatz's Conjecture (E16 The 3x+1 problem)

1. This is very famous unsolved problem.
For n∈N, define f(n) as,

3n+1 (n is odd)
n/2 (n is even).
Then any n (seems to) converge to the sequence 4 - 2 - 1.

2. I suggest how to construct the apparantly convergent equation like 4n+1.
(4n+1 -- (12n+3)+1 -- 3n+1,   (3n+1) < (4n+1) )
By using this method, we can reduce the searching range.
The ratio are shown here.

3. I also consider the relation between n and max(n) (largest value of f(n)).
The table is here. It seems to be that there exists constant s (almost equal to 2) such that

max(n) ≤ n2+ε
If it is proven, it would be the upper bound of f(n) for arbitrary n,
and we can prove the original problem (but it seems to be very difficult).

For further results, please see Tomas Oliveira and Silva's web site.
3x+1 conjecture verification results

#### Chapter 8 : Continued Fraction and Pell's Equation

1. Diophantine equation x2-ny2=±1 is called Pell equation.
Sometimes x and y becomes large for some n. For example,

2275282 - 103 * 224192 = 1
88901822 - 109 * 8515252 = -1

2. We can solve Pell's Equation using continued fraction expansion.
A program is here. (this is not an unsolved problem).

#### Chapter 9 : Amicable Numbers (B4 Amicable numbers, B5 Quasi-amicable or betrothed numbers)

1. Searching the following number's pairs up to 1010.

Amicable numbers
Quasi-amicable numbers
Augmented amicable numbers.

#### Chapter 10 : Congruent Numbers (D27 Congruent numbers)

1. (A few contents are still written in Japanese)

2. If n∈N is the value of area of right-angled triangle and the values of each sides of triangle are in Q, then n is called as congruent number or congruum.
How to construct the solution of n is not known.

3. I showed some construction method for the solutions of congruum.

4. The solutions of congruum up to 1000 are here and locate here.

#### Chapter 11 : Number Theoretic Algorithms

1. (A few contents are still written in Japanese)

2. Introduction of various number theoretic algorithms.

3. Following topics are rarely seen in the elementary course, I think.

1. The coefficients of Taylor expansion of Tan z.
2. How to calculate partition numbers by not a recursive way.
3. How to calculate Lucas sequence.
4. Bernoulli numbers and Euler numbers (B45 Euler numbers)
4. Wieferich prime (A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k.2n+1)

5. an≡c (mod n) (F10 Residues of powers of two)

6. Hilbert class polynomial, Weber class polynomial

#### Chapter 12 : Integer Factorization Algorithms

1. (A few contents are still written in Japanese)

2. Following algorithms are mentioned, and also shown the sample programs.

1. brute force method
2. rho method
3. p-1 method
4. p+1 method
5. continued fraction method
6. multiple polynomial quadratic sieve method
7. elliptic curve method

#### Appendix 1 : WIFC (World Integer Factorization Center)

including the factors of following problem;

• A2 Primes connected with factorials
• A3 Mersenne primes. Repunits. Fermat numbers. Primes of shape k.2^n+1
• B20 Cullen and Woodall numbers
• B43 Alternating sums of factorials
• B44 Sums of factorials
• B45 Euler numbers
• Wolsten primes which is introduced in the book, "Prime Numbers: The Most Mysterious Figures in Math" 