I have mention Brocard’s problem in a previous post
https://kevinjdavis2013.wordpress.com/2014/12/30/brocards-problem-1897/
In this I noted the ‘proof’, or at least the scoping of the problem as only having a finite number of solutions, if the 
In the three years since my post, computational studies have continued with ever larger numbers but still no counter examples have been found.
A rich seam of research has also studied variations of the Brocard – Ramanujan equation
for example by replacing the factorial 
I have been looking at an additive analogue of the Brocard – Ramanujan equation.
The factorial of 
The Brocard – Ramanujan equation as an additive analogue is
Positive integer solutions of this variant can be found, and there are many(!); 
By noting that
it is simple to show that for the positive integer solutions of this Brocard – Ramanujan analogue, 
Do other number theory conjectures yield similarly simple proofs on rewriting in an additive or multiplicative analogue form?
More to follow …
Calendar from Kom Ombo (my photograph from Egypt)
Saqqara (my photograph from Egypt) 
(with the induced topology) as discussed in the previous two posts.
in the variables 
and 
defines a plane curve affine variety 
. Below is the variety for the plane elliptic curve 
.
in 
is an affine variety. It is the intersection of an ellipse 
and a hyperbola 
.
. In 
these are; spheres, ellipsoids, paraboloids, and hyperboloids in 
.
and an elliptic paraboloid 
.
and a hyperboliod of two sheets 
.
and 
that is a `figure of eight’ curve in 
in 
and the surface 
in 
are irreducible hyper-surfaces. These are visualised below; the first has components that are connected in Euclidean topology, the second has five components that meet at four singular points.
, is defined as
.
, integer mutilples of 
, is 
. The radical ideal of 
is 
is an algebraically closed field with ![A=k[x_1,\dots ,x_n]](https://s0.wp.com/latex.php?latex=A%3Dk%5Bx_1%2C%5Cdots+%2Cx_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
and 
a polynomial that vanishes at all points of 
, then 
for some integer 
.
are subsets of 
, then 
.
are subsets of 
.
, 
of 
.
, 
. This is a consequence of Definition 2.2.
, 
, the closure of 
.
of a commutative ring 
is prime if it has the following properties:
such that 
, then 
or 
,
, the maximal ideals are the principle ideals generated by a prime number.
correspondence ![\{points\; in \; {A_k}^n\}\Leftrightarrow \{ maximal \; ideals \; of \; k[x_1, \dots, x_n]\}](https://s0.wp.com/latex.php?latex=%5C%7Bpoints%5C%3B+in+%5C%3B+%7BA_k%7D%5En%5C%7D%5CLeftrightarrow+%5C%7B+maximal+%5C%3B+ideals+%5C%3B+of+%5C%3B+k%5Bx_1%2C+%5Cdots%2C+x_n%5D%5C%7D&bg=3c3c3c&fg=999999&s=0&c=20201002)
.![k[x_1,\dots ,x_n]](https://s0.wp.com/latex.php?latex=k%5Bx_1%2C%5Cdots+%2Cx_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
has a zero in 
be a finite algebraic set. 
is generated by the function 
, and 
so 
is the inverse of 
.
is an affine variety iff its ideal ![I(X)=k[x_1, \dots, x_n]](https://s0.wp.com/latex.php?latex=I%28X%29%3Dk%5Bx_1%2C+%5Cdots%2C+x_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
is a prime ideal.
has a dimension of 
as single points are the only irreducible closed subsets of 
is an irreducible polynomial in ![A=k[x_1, \dots , x_n]](https://s0.wp.com/latex.php?latex=A%3Dk%5Bx_1%2C+%5Cdots+%2C+x_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
, the affine variety 
is called a surface if 
and a hyper-surface of 
.
is said to be Noetherian if it satisfies a descending chain condition for closed subsets. For any sequence 
of closed subsets there is an integer 
such that 
. Another way of saying this is that the closed subsets are stationary.
with a mapping 
. Then 
is inclusion-reversing iff for every pair of sets 
such that 
.
and 
.
maximum ideals in a polynomial ring. (Corollary 2.1)
that does not have a solution in the real numbers.
is a set of all 
is called a point and for 
, then 
are called the coordinates of 
.![A=k[x_1,\dots,x_n]](https://s0.wp.com/latex.php?latex=A%3Dk%5Bx_1%2C%5Cdots%2Cx_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
be the polynomial ring in 
and 
. If 
is a polynomial the set of zeros of f are 
. Generally, if 
is any irreducible subset of 
So an affine algebraic variety is the set of common zeros of a collection of polynomials.
such that 
. The ideal is generated by 
. e.g the ideals 
of the ring of integers 
. We note that a function 
although the ideals are, of course, different. If 
is and ideal of 
.
there exists an 
i.e. the ascending chain of ideals terminates. 
. Thus 
can be expressed as the common zero’s of the finite set of polynomials 
such that 
.
, 3) whenever two or more sets are in 
. A set 
is an open set then its complement (
is called a closed set. Note we can define a topology on a set 
with subsets 
comprise a topology and 
. Here the point 
and 
with 
. Let ![A=k[x_1]](https://s0.wp.com/latex.php?latex=A%3Dk%5Bx_1%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
be the polynomial ring in one variable over 
and 
. Every ideal in 
, with all 
. Then 
. Thus the algebraic sets in 
). The open sets are the empty sets and the complements of the finite subsets.![k[x_1, \dots, x_n]](https://s0.wp.com/latex.php?latex=k%5Bx_1%2C+%5Cdots%2C+x_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
. The operation ![k[x_1, \dots ,x_n]](https://s0.wp.com/latex.php?latex=k%5Bx_1%2C+%5Cdots+%2Cx_n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
or an ideal to an algebraic set. With the following definition we establish a two way correspondence.
we call the ideal ![I(X):=\lbrace f \in k[x_1, \dots, x_n] ; f(P)=0 \; for \; all \; P \in X \rbrace \subset k[x_1,\dots, x_n]](https://s0.wp.com/latex.php?latex=I%28X%29%3A%3D%5Clbrace+f+%5Cin+k%5Bx_1%2C+%5Cdots%2C+x_n%5D+%3B+f%28P%29%3D0+%5C%3B+for+%5C%3B+all+%5C%3B+P+%5Cin+X+%5Crbrace+%5Csubset+k%5Bx_1%2C%5Cdots%2C+x_n%5D+&bg=3c3c3c&fg=999999&s=0&c=20201002)
Hence we have defined the following two way correspondence; we have an operation 
. If 
, if 
, otherwise there are two values for 
.
around the complex origin, the square root of this 
which gives opposite values at 
and 
. If in 
we travel a path around any of the points 
we move from one of the copies of the complex plain to the other.![[1,2],\dots,[2n-1,2n]](https://s0.wp.com/latex.php?latex=%5B1%2C2%5D%2C%5Cdots%2C%5B2n-1%2C2n%5D&bg=3c3c3c&fg=999999&s=0&c=20201002)
of 
meet: this results in a compact surface with 
handles. Such objects are called surfaces of genus 
, which results in a surface of genus 
kevinjdavis2013 