LOG#158. Ramanujan’s equation.

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?

(1) $\begin{equation*} 2^n-7=x^2 \end{equation*}$

This equation can be rewritten as follows

(2) $\begin{equation*} 2^{p+3}-7=x^2=(2n-1)^2 \end{equation*}$

The full set of solutions in terms of the pair (p,n) can be found to be:

(3) $\begin{equation*} S=\left[(0,0),(1,2),(2,3),(4,6),(12,91)\right] \end{equation*}$

If we put

$n=(x+1)/2$

the solutions are obvious and, the substitution allows to write

$2^{p+3}-7=x^2=(2n-1)^2$

as the transformed equation

$2^p-1=(n-1)n/2$

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:

(4) $\begin{equation*}2^p-1=(n-1)n/2=\begin{pmatrix} n\\ 2\end{pmatrix}=\sum_{k=1}^{n-1}k\end{equation*}$

This equation has the next set of solutions:

1st.

$2^0-1=(1-1)1/2=\sum_{k=1}^{1-1}k=0$

2nd.

$2^1-1=(2-1)2/2=\sum_{k=1}^{2-1}k=1$

3rd.

$2^2-1=(3-1)3/2=\sum_{k=1}^{3-1}k=3$

4th.

$2^4-1=(6-1)6/2=\sum_{k=1}^{6-1}k=15$

5th.

$2^{12}-1=(91-1)91/2=\sum_{k=1}^{91-1}k=4095$

These relationships are connected (secretly) with the Clifford algebras and orthogonal groups in the following way.

1st. Trivial and nullity

$C(0)\rightarrow O(0)$

2nd. Trivial as well

$C(1)\rightarrow O(1)$

3rd. First non trivial result (known to mathematical and physicists)

$C(2)\rightarrow O(3)$

4th. Non trivial case (a little bit unknown case, linking rotations in six dimensions with the Clifford algebra of spacetime):

$C(4)\rightarrow O(6)$

5th. Mysterious case (not easily found in the literature), highly non trivial and VERY unknown to physicists and mathematicians (to my knowledge)

$C(12)\rightarrow O(91)$

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!

PS:

$\chemfig{*6(-=-*6(-(-*6(=-=-=-))-*6(=-=-=-))=-=)}$

$\chemfig{*5(-=--=)}$

$\chemfig{**5(-----)}$

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LOG#158. Ramanujan’s equation.

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