In this paper we extend a notion of Cassini determinant to recently introduced hyperfibonacci sequences. We find Q-matrix for the r-th generation hyperfibonacci numbers and prove an explicit expression of the Cassini determinant for these sequences.

An known elementary proof of existence of infinitely many solutions (and a procedure of their construction) to the Prouhet-Tarry-Escott problem is given together with its connection to the Prouhet-Thue-Morse sequence.

No abstract

In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for n = 3, 4 and for general n only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D.D -oković). Interestingly the conjecture C2 (and also stronger C3) is not yet proven even for arbitrary four points in a plane. So far we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra.We have also proposed a strengthening of conjecture C3 for configurations of four points (Four Points Conjectures).For almost collinear configurations (with all but one point on a line) we propose several new conjectures (some for symmetric functions) which imply C2 and C3. By using computations with multi-Schur functions we can do verifications up to n = 9 of our conjectures. We can also verify stronger conjecture of D -oković which imply C2 for his nonplanar configurations with dihedral symmetry.Finally we mention that by minimizing a geometrically defined energy, figuring in these conjectures, one gets a connection to some complicated physical theories, such as Skyrmions and Fullerenes.

In 2001 Sir M. F. Atiyah formulated a conjecture C1 and later with P. Sutcliffe two stronger conjectures C2 and C3. These conjectures, inspired by physics (spin-statistics theorem of quantum mechanics), are geometrically defined for any configuration of points in the Euclidean three space. The conjecture C1 is proved for n = 3, 4 and for general n only for some special configurations (M. F. Atiyah, M. Eastwood and P. Norbury, D.D -oković). Interestingly the conjecture C2 (and also stronger C3) is not yet proven even for arbitrary four points in a plane. So far we have verified the conjectures C2 and C3 for parallelograms, cyclic quadrilaterals and some infinite families of tetrahedra.We have also proposed a strengthening of conjecture C3 for configurations of four points (Four Points Conjectures).For almost collinear configurations (with all but one point on a line) we propose several new conjectures (some for symmetric functions) which imply C2 and C3. By using computations with multi-Schur functions we can do verifications up to n = 9 of our conjectures. We can also verify stronger conjecture of D -oković which imply C2 for his nonplanar configurations with dihedral symmetry.Finally we mention that by minimizing a geometrically defined energy, figuring in these conjectures, one gets a connection to some complicated physical theories, such as Skyrmions and Fullerenes.