Tünde Lality-Kovács JSPS Postdoctoral Fellow

Nihon University, Tokyo

Located in Central Europe

Area: 93000 km² Capital: Budapest Landlocked

country

Hévíz: largest thermal lake in the world

Balaton: Central Europe’s largest lake

More than 50% of the land is plain

Kékes: Hungary’s highest mountain 1014 m

Population: 9.940.000 people

Population density: 107 people/ km²

Main industy: mining textiles

agriculture: wheat, corn, potato; livestock (cattle, pig, poultry)

Rubik’s cube Ballpoint pen Holography Helicopter Plasma TV C-vitamin Refrigerator Match Telephone exchange etc ...

Important ingredients: paprika, garlic, onion, meat, potato

Famous food: goulash, fishermen’s soup, sausages, cakes

Famous beverages: palinka, wine, unicum

Széchenyi bath

Parliament

Buda castle

porcelain

sausage

Túró rudi

Kalocsa embroidery

Erdős Pál (1913-1996) 1525 papers 511 co-authors Erdős-number: describes

a person’s degree of separation from Erdős Combinatorics, graph theory, number theory,

classical analysis, approximation theory, set theory, probability theory

The book with title 博士の愛した数式 was a best-seller in Japan

The book is a novel but the story is based on the life of Erdős

Lajos Hajdu he was my main

inspiration to become a mathematician

Gauss, one of the greatest mathematicians said:

„Mathematics is the queen of sciences and

number theory is the queen of mathematics.”

Number theory is the study of the set of positive whole numbers

1, 2, 3, 4, 5, 6, 7, . . . , which are often called the set of natural numbers. Since ancient times, people have separated the

natural numbers into a variety of different types. Here are some familiar examples: odd: 1, 3, 5, 7, 9, 11, ... even: 2, 4, 6, 8, 10, ... square: 1, 4, 9, 16, 25, 36, ... cube: 1, 8, 27, 64, 125, ... prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... composite: 4, 6, 8, 9, 10, 12, 14, 15, 16, ...

The main goal: discover interesting, unexpected relationships between different sorts of numbers; prove that these relationships are true.

Number theory is partly experimental and partly theoretical.

The experimental part normally comes first; it leads to questions and suggests ways to answer them.

The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions.

1. Accumulate data, usually numerical. 2. Examine the data and try to find patterns

and relationships. 3. Formulate conjectures (i.e. guesses) that

explain the patterns and relationships. These are frequently given by formulas.

4. Test your conjectures by collecting additional data and checking whether the new information fits your conjectures.

5. Prove that your conjectures are correct.

Let’s consider the sequence of perfect powers! perfect power: a perfect power is a positive

integer that can be expressed as an integer power of another positive integer;

or more formally, n is a perfect power if

there exist natural numbers m > 1, and k > 1 such that mk = n.

What are the perfect powers between 1 and 100?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

What are the perfect powers between 1 and 100?

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

What are the perfect powers between 1 and 100?

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100

4=22 32=25 8=23 36=62 9=32 49=72

16=24=42 64=26=43=82 25=52 81=34=92 27=33 100=102

4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 What kind of questions can we ask about them? For

example: What are the differences between the consecutive

perfect powers? What is the smallest difference between two

perfect power? Is there any perfect power other than 8 and 9

whose difference is one?

Conjecture: 8 and 9 are the only consecutive perfect

powers; in other words,

32-23=1 is the only non-trivial solution to the equation

xp-yq=1. This is called Catalan’s conjecture.

Compute perfect powers between 1 and 1000 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100,

121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, 512, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1000

Is our conjecture true for this larger set? Yes!

This is undoubtedly the hardest part of the 5-step process.

Actually, it took more than 150 years to find a

proof for Catalan’s conjecture. Showing the proof is beyond the scope of

today’s lecture.

As an exercise try to prove the special case p=q=2!

The case „p=2 and q=3”: Euler, 1738: The only nonzero rational solutions to the Diophantine equation

x2=y3–1 are given by x=±3 and y=2.

The case „q=2”: Lebesque, 1850: For any exponent p≥2, the

Diophantine equation xp=y2+1

has no solution in nonzero integers x,y. The case „p=2”: Ko Chao, 1965: Let q≥5 be prime. Then the

Diophantine equation x2–yq=1

has no solution in nonzero integers x,y.

1844: Catalan stated his conjecture

1976: Tijdeman showed that there exist only finitely many pairs of consecutive perfect powers; his proof is based on the theory of linear forms in logarithms; unfortunately his bound is astronomical

1999: refinement of Tijdeman’s work: Catalan’s conjecture is true for p,q>7.78∙1016

2003: Mihăilescu proved the conjecture; his proof does not rely on Tijdeman’s work or on computer calculations

Runge’s method Cassels’ theorem Cyclotomic fields Obstruction group Stickelberger ideal Double Wieferich criterion Semisimple group rings The density theorem Thaine’s theorem etc...

Budapest with the river Danube

Hungary, a small country in central Europe��Catalan’s conjecture, a famous problem of number theoryHungary’s geographyHungary’s geographyDemography, industryInventions by HungariansCuisinearchitecturenatureSpecial �productsPlease come and visit Hungary!Hungary – �mathematicsHungary – �mathematicsUniversity of debrecenMy professorNumber theoryWhat is number theory?The goal of number theoryHow to solve a number theoretical problem?Let’s see a concrete exampleStep 1 – accumulate dataStep 1 – accumulate dataStep 1 – accumulate dataStep 2 – examine the dataStep 3 – formulate conjectures, usually by formulasStep 4 – test your conjecture Step 5 – prove that the conjecture is correctSome special cases of catalan’s conjectureSome special cases of catalan’s conjectureHistory of Catalan’s conjectureHistory of Catalan’s conjectureHistory of Catalan’s conjectureTheories behind the proofThank you very much for your attention!