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In this illusion, the vertical lines in the above figure are parallel, but appear to be tilted at an angle. In 1860, F. Zöllner sent his discovery in a letter to physicist ...

If 1<=b<a and (a,b)=1 (i.e., a and b are relatively prime), then a^n-b^n has at least one primitive prime factor with the following two possible exceptions: 1. 2^6-1^6. 2. ...

A graph whose nodes are sequences of symbols from some alphabet and whose edges indicate the sequences which might overlap. The above figures show the first few n-dimensional ...

Graphs corresponding to closed walks of length k are known as k-cyclic graphs, or C_k-graphs for short. C_k-graphs are connected by definition. The numbers of C_k-graphs for ...

A p-adic integer is a p-adic number of the form sum_(k=m)^(infty)a_kp^k, where m>=0, a_k are integers, and p is prime. It is sufficient to take a_k in the set {0,1,...,p-1}. ...

Clausen's _4F_3 identity _4F_3(a,b,c,d; e,f,g;1)=((2a)_(|d|)(a+b)_(|d|)(2b)_(|d|))/((2a+2b)_(|d|)a_(|d|)b_(|d|)), (1) holds for a+b+c-d=1/2, e=a+b+1/2, a+f=d+1=b+g, where d a ...

Pairs of partitions for a single number whose Ferrers diagrams transform into each other when reflected about the line y=-x, with the coordinates of the upper left dot taken ...

Let a Gram point g_n be called "good" if (-1)^nZ(g_n)>0, and "bad" otherwise (Rosser et al. 1969; Edwards 2001, p. 180). Then the interval between two consecutive good Gram ...

An extension of a group H by a group N is a group G with a normal subgroup M such that M=N and G/M=H. This information can be encoded into a short exact sequence of groups ...

The Hofstadter ellipses are a family of triangle ellipses introduced by P. Moses in February 2005. The Hofstadter ellipse E(r) for parameter 0<r<1 is defined by the trilinear ...