## ag.algebraic geometry – divisibility of a divisor

To let $$X$$ to be a smooth complex projective curve and $$f colon X to Y$$ an étale Galois cover, whose Galois group $$G$$ is finite and tidy $$r$$, For all $$g in G$$, define $$Delta_g = {(x, , g cdot x) ; | ; x in X } subset X times X.$$ Then everyone $$Delta_g$$ is a smooth divisor isomorphic to the diagonal $$Delta = Delta_1$$,

In addition, the fact is that $$f$$ is étale implied $$Delta_g cap Delta_h = emptyset$$ if $$g neq h$$so that the reducible divider $$D = sum_ {g in G} Delta_g$$ is smooth (note that $$D$$ is the fiber product $$X times_Y X$$).

question, is $$Delta$$ $$r$$divisible into $$mathrm {Pic} (X times X)$$?

## Number theory – divisibility check in different modules

So I am working on modular arithmetic and have encountered the following problem:

Accept that $$a cdot b mod m = c$$ and I know that $$c mod n = d$$, Can I still test for divisibility? $$c$$ With $$d$$? A quick check tells me if that works $$d mod n = 0$$ does not work – but is this review not possible or is there a general way to find out $$c | a$$? Maybe, if $$m$$ and $$n$$ are primes?

Many Thanks!

## Number theory – check of divisibility with minimal bits

Suppose we get an infinite stream of integers. $$x_1, x_2, …$$,

a) Show that we can calculate whether the sum of all integers seen so far is divisible by a fixed integer $$N$$ With $$O (log N)$$ Bits of memory.

b) Leave $$N$$ an arbitrary number, and let's assume we are given $$N$$Hauptfaktorisierung: $$N = p_1 ^ {k_1} p_2 ^ {k_2} … p_r ^ {k_r}$$, How would you check if $$N$$ divides the product of all integers $$x_i$$ So far, use as little memory as possible? Make note of the number of bits used in $$k_1, …, k_r$$,

For part a) we know this for every prime number $$p ne 2, 5$$There is an integer
$$r$$ so that to see if $$p$$ divides a decimal number $$n$$we interrupt $$n$$ in $$r$$-Tuple of decimal places, add these $$r$$tuple and check if the sum is divisible by $$p$$, But $$N$$ is a fixed integer and not necessarily a prime. Is there a way to join the above sentence with any number?

For part b) for the product of $$x_i$$ (call $$y$$) is divisible by $$N$$, then $$y$$ must be divisible by everyone $$p_i ^ {k_i}$$ (call $$a_i$$, Since we get the prime factorization, we can only check if $$y$$ is divisible by $$N$$ by sharing $$y$$ of each $$a_i$$ and stop, if it fails, right? Would this lead to the use $$k_1 * … * k_r$$ Bits?

Am I at least on the right track or am I completely wrong? Any help that understands this problem would be enormously helpful. Many Thanks.

## nt.number theory – integer partitions under divisibility constraint

Consider integer partitions of $$x in mathbb {N}$$ of the size $$k$$ on the condition that the partition elements are different and the ratio of an element to each smaller element is a natural number.

example $$f (x, k)$$:

• $$f (17.3) = {1, 4, 12 }$$
• $$f (14,3) = {2, 4, 8 }$$
• $$f (101,4) = {1, 2, 14, 84 }$$
• $$f (4,3) = emptyset$$

As @Henrik has pointed out, one limitation can be expressed as follows: $$x = a_1 + a_1 a_2 + ldots + a_1 cdots a_k$$ and then factor $$x$$, $$x-a_1$$, $$x-a_1 a_2$$etc. to find candidates for successive ones $$a_i$$but because there are usually several options for each successive one $$a_i$$Maybe you have to use a sophisticated search (with backtracking) or a variant of linear programming.

Were these partitions (or series) examined? Is there an efficient algorithm or method to find it? $$x$$ and $$k$$?