Arithmetic Properties Of Polynomial Coefficients

For any integers n≥1,m≥2,k1≥0,…,km≥0,k1+k2+…+km=n, define it is easy to see that(k1,k2,…,kmn)is the coefficient of the term x1k1x2k2x…mkm in multinomial(x1+x2+…+xm)n,hence,(k1,k2n…,km)is named multinomial coefficeint.When n=2,(k1,k2n)is the ordinary binomial coefficient.There are various researches on binomial coefficients and multinomial co-efficients,and there are a lot of pages talking about binomial coefficients and multinomial coefficients,e.g. in Guy’s Unsolved Problems in Number Theory and Dickson’s History of the Theory of Numbers,these two books are well-known among number theorists.The research on binomial coefficients and multinomial coefficients could be classified into three subject:the divisibilities of multinomial coefficients,the congruences about multinomial coefficients and the diophantine equations related to multinomial coefficients.In this thesis, we mainly introduce the following three kind of sums of multi-nomial coefficients Our major results are:1.In the second chapter, we focus on the internal structures about the multi-nomial sums Sn,mr, and we show that Sn,mr could be divided by m. 2. In the third chapter, we give several congruences about multinomial coeffi-cients, and we discuss the great common divisor of Sn,mr and Tn,mr, basing on the congruences and the results of the second chapter3. In the fourth chapter, we generalize McIntosh’s theories of Lucas property, from which we give a relation between multiple Lucas properties and the original Lucas properties. Later, we prove that both Sn,mr and Tn,mr have Lucas properties. We also difine a partial high order Lucas property, and we prove that Sn,mr, Tn,mr, and Apery numbers have partial2order Lucas properties.4. In the finial chapter, we discuss a kind of diophantine equations about multinomial coefficients, and we proved that the multinomial coefficients are almost not integer power.