Hi, everyone! I am back, again! And I have some new toys in order to post faster (new powerful plugin). Topic today: Ramanujan!
Why Ramanujan liked the next equation?
This equation can be rewritten as follows
The full set of solutions in terms of the pair (p,n) can be found to be:
If we put
the solutions are obvious and, the substitution allows to write
as the transformed equation
Therefore, the Ramanujan’s “seven equation” is deeply related with the following problem:
“(…)Find and calculate the values of the pair (p,n) such as a Mersenne prime number is a triangular number(…)”
Currently, we do know that the above equation has the previously quoted 5 solutions. Curiously, Ramanujan also loved the number 5 (not only the number 7) since he liked the golden ration, continuous fractions and related topics. In fact, the Rogers-Ramanujan identities are important in Mathematics and Physics.
In particular, we also observe:
This equation has the next set of solutions:
These relationships are connected (secretly) with the Clifford algebras and orthogonal groups in the following way.
1st. Trivial and nullity
2nd. Trivial as well
3rd. First non trivial result (known to mathematical and physicists)
4th. Non trivial case (a little bit unknown case, linking rotations in six dimensions with the Clifford algebra of spacetime):
5th. Mysterious case (not easily found in the literature), highly non trivial and VERY unknown to physicists and mathematicians (to my knowledge)
Two recent news I liked very much:
1) Pentagraphene: a new graphene-like structure and material.
2) A new type of bond beyond those you know from high school (ionic, covalent and metallic): THE VIBRATIONAL BOND (found in bromine-muonium systems Br-Mu-Br).
See you later, I wish!