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Demonstratio insignis theorematis numerici circa uncias potestatum binomialium
This paper is about the product z^q/(1 - z)^(q + 1)(1 + (z/(1 - z)))^p, Euler gives the Talylor-Series and takes a closer look at the coefficient.
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Theorematum quorundam arithmeticorum demonstrationes
Euler proves that the sum of two 4th powers can't be a 4th power and that the difference of two distinct non-zero 4th powers can't be a 4th power and Fermat's theorem that the equation x(x+1)/2=y^4Expand
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Nova demonstratio, quod evolutio potestatum binomii Newtoniana etiam pro exponentibus fractis valeat
Here Euler notes the recursive relation for the general binomial coefficients, by assuming that (1+x)^a can be expanded in a power series.
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Speculationes super formula integrali {\int} (x^ndx)/{\surd}(aa-2bx+cxx), ubi simul egregiae observationes circa fractiones continuas occurrunt
Euler evaluates the integrals in the title and recognizes a recursion between them, which he then uses to give continued fractions for the log and arctan. The paper is translated from Euler's LatinExpand
Disquitiones analyticae super evolutione potestatis trinomialis (1+x+xx)^n
Euler investigates the Taylorseries of (1+x+xx)^n and uses the results to evaluates some integrals which are today often proved with the calculus of residues.
Commentatio in fractionem continuam, qua illustris La Grange potestates binomiales expressit
Euler gives a continued fraction representation of (1 + x)n. involving 1,3,5,7,... and n^2-1,n^2-4,n^3-9,... and squares of z, for x=2y and y=z/(1-z). He evaluates this continued fraction at z=tExpand
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Observationes generales circa series, quarum termini secundum sinus vel cosinus angulorum multiplorum progrediuntur
This paper, along with E592 and E636, seems to consider the binomial expansion (1+z)^n in the case where z is complex. Euler even gives the sums of divergent series. The paper is translated fromExpand