Abstract. I revisit an automated proof of Andrews’ pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews’ result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater ...

the edge szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph g, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. in this paper, the edge szeged index of one-pentagonal carbon nanocone cnc5[n] is computed for the first ...

Pentagonal conjugates of tryptophane zipper-forming peptide (CKTWTWTE) with a pentaazacyclopentadecane core (Pentagonal-Gly-Trpzip and Pentagonal-Ala-Trpzip) were synthesized and their self-assembling behaviors were investigated in water. Pentagonal-Gly-Trpzip self-assembled into nanofibers with the width of about 5 nm in neutral water (pH 7) via formation of tryptophane zipper, which irreversi...

Resting cyst and motile thecate cell stages of three heterotrophic Protoperidinium oblongum, Protoperidinium sp and Protoperidinium claudicans were assessed. Cysts were isolated from sediment collected from southeast coast of Iran. Individual live cysts were incubated under optimal conditions for germination. Results showed that Protoperidinium oblongum cysts were pentagonal smooth walled and t...

Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number in his notebooks theorem around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty...