We solve the difference equation with linear coefficients by the Momentenansatz to obtain explicit formulas for orthogonal polynomials.
We give another proof for [ sum_{n=1}^{infty}frac{1}{n^2}=frac{pi^2}{6} ] that basically follows from the theory of difference equations.
We translate the paper De Integralibus quibusdam definitis, seriebusque infinitis
Euler presents a third proof of the Fermat theorem, the one that lets us call it the Euler-Fermat theorem. This seems to be the proof that Euler likes best. He also proves that the smallest power x^n that, when divided by a numer N, prime to x, and t
hat leaves a remainder of 1, is equal to the number of parts of N that are prime to n, that is to say, the number of distinct aliquot parts of N. The translation is presnted from Eulers Latin original into German.
Euler proves that the sum of two 4th powers cant be a 4th power and that the difference of two distinct non-zero 4th powers cant be a 4th power and Fermats theorem that the equation x(x+1)/2=y^4 can only be solved in integers if x=1 and the final the
orem y^3+1=x^2 can only be solves for x=3 and y=2 in integers. The paper is translated from Eulers Latin original into German.
Euler gives a long introduction, giving all the arguments for and against the use of divergent series in calculus and then gives his own definition of the sum of a diverging series. Then in the second half of this paper he evaluates the the 1-1+2-6+2
4-120+720-... on several ways and gets the sum 0.5963473621372. The paper is translated from Eulers Latin original into German.
This paper does exactly what the title says it does. It expands the given series to arrive at the familiar pentagonal number expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Eulers Latin original into German.
