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Schweizer Jugend Forscht Arbeit: Oliver Knill, 1981, Anschauliche Zahlentheorie

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As a high school student, I pursued some research in additive number theory. Equipped with a PC, I followed a methodology which now is known as experimental mathematics: I tried to find mathematical results empirically using experiments and then tried to prove them with the methods I had at hand. Proving statements was sometimes possible with primitive combinatorial tools. But often, like for the pentagonal theorem (the result I'm most proud of to have discovered during my experiments), a proof was out of range for me. The only literature, I had consulted during my research was an one-volume encyclopaedia on mathematics, in which the partition function p(n) was mentioned. While wrapping up, I visited the fantastic library at ETH and found a book on additive number theory. While it was too advanced for me to read, I realized that most of the material I had discovered was already known. And that much more was already known. Later in college, I learned about Young diagrams and the wonderful tool of generating functions. Having used geometric methods myself using Cuisinaire material, the Young diagram method was not too surprising. But the idea of using generating functions to tackle combinatorial problems was a shock to me and I would go to the other extreme during my college years: I would essentially stop doing research and fill my head with all kind of mathematics. The participation in the contest "Swiss Youth does Research" added an other valuable experience: I saw what differences there are in "experts" and how a referee process can look like. While some people could read and understand what I did, (the expert in the national contest was such a person as well as my high school teacher to whom I had dedicated the research), the expert in the regional contest did not even understand the question which was addressed, nor could he place the results in which field of mathematics it belonged to. Here is a summary of the main results described in the abstract article. The starting question was to find out, in how many ways a certain amount of money could be changed. In Switzerland, where 5 cent is the smallest unit, the other units for which coins or bills exist are A={1,2,4,10,20,49,100,200,400,1000,2000,4000,10000,20000}. Let P(A,n) denote the number of ordered sequences (a1,...,ak) such that a1 + ...+ ak =n and P(A,n) = P(A,n-a1) + P(A,n-a2) + ... + P(A,a-ak). For the coin problem there are P(A,20)=48'008 possibilities to change 1 Swiss Frank. If we ask the same but that the order of the sequence should not matter, we obtain a different answer. Let p(A,n) denote the number of different sums n=a1+...+ak we can build with element in A. Let Aj= {aj,...,ak}, we have the recurion p(A,n) = suma in A p(Ai,n-ai). For the Swiss currency exchange problem, one has p(A,20)=50. A geometrical problem in additive number theory is the question, how many triangles with perimeter n and integer sides there are. For example, for n=9, there are 3 triangles. The problem is to find all partitions of n=a+b+c where a,b,c are positive numbers and the sum of two numbers is larger then the third. We found an explicit formula for this combinatorial problem. Can one find algorithms to compute p(n) efficiently. The function p(n) is the number of ways to write n as a sum of positive integers. I found the following formulas: If s(k) is the sum of all divisors of n, then

p(n) = sumi=1n p( (i,i+1,...,n),n-i)
p(n) = sumi=1n s(k) p(n-k)/n
p(n) = sumk (-1)(k-1)p(n-(3k2-k)/2) + p(n-(3k2+k)/2)

I was able to prove the first two formulas. The last formula is the famous pentagonal theorem of Euler. There were further topics, like problems in game theory or geometric problems like in how many ways one can write a cube of size n x n x n as a union of smaller cubes.

Discussion with Dr. Hans Giger, 1981, who was an excellent referee. Giger did his PhD with Willy Scherrer and published also with Hugo Hadwiger.

Put online: 6/22/2006.

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