Number theory – divisibility properties of Pisano periods

To let $ (F_n) $ the Fibonacci sequence and $ pi (m) $ the pisano time of $ m $ (ie the smallest period of $ F_n pmod {m} $). There are many proven results over $ pi (m) $, For example, this is known $ pi (p ^ a) mid p ^ {a-1} pi (p) $, for any prime $ p $ and an integer $ a geq 1 $, it is supposed that $ pi (p ^ a) = p ^ {a-1} pi (p) $,

My question is a weak version of this assumption. For example, I want to prove that $ m ^ 2 mid pi (m) $ for all positive integers $ m $,

Since for $ m = p_1 ^ {a_1} cdots p_k ^ {a_k} $it holds $ pi (m) = lcm ( pi (p_1 ^ {a_1}), lpoints, pi (p_k ^ {a_k})) $Then it is enough to prove that
[
p^amid pi(p^{2a}),
]
For every prime $ p $ and an integer $ a geq 1 $, There's the guess $ pi (p ^ a) = p ^ {2a-1} pi (p) $We have some freedom, though $ a <2a-1 $ (ie. $ a> 1 $). Could someone help me?

Actually, even in the case of $ m mid pi (m ^ 3) $ would be helpful. Thank you in advance!

Divisibility of the sum of multinomials

To let $ n, m $ and $ t $ be positive integers. Define the multi-family sequences
$$ S (n, m, t) = sum_ {k_1 + cdots + k_n = m} binom {m} {k_1, dots, k_n} ^ t $$
where the sum runs over non-negative integers $ k_1, dots, k_n $, These numbers refer to average distances (from the origin) of random, uniform step-steps in the plane.

QUESTION. Is it always like that? $ n $ splits $ S (n, m, t) $?

Watch that $ S (n, m, 1) = n ^ m $,

nt.number theory – Systems of $ n $ divisibility conditions on $ n $ Prim variables

Consider a system of $ n $ Divisibility conditions on $ n $ Prim variables:
$$ p_i | a_ {i, 1} p_1 + dotsc + a_ {i, n} p_n, ; ; ; ; ; ; 1 leq i leq n, $$
from where $ a_ {i, j} $ are integers. How many solutions are there at all? $ p_i $ Prime and in one area $[N_0,N_1]$? (We can guess that $ N_0 $ is a bit bigger than all $ a_ {i, j} $In other words, under what conditions are there very few solutions?


Some thoughts: If $ lbrack N_0, N_1 rbrack $ is dyadic (ie the shape) $ lbrack N, 2N rbrack $) then the divisibility conditions can be replaced by a system of equations $ c_i p_i = a_ {i, 1} p_1 + dotsc + a_ {i, n} p_n $, $ c_i $ limited. That's a system of $ n $ Equations in $ n $ Variables and should therefore generally have no non-trivial solutions (which means that there are no solutions, such as $ 0 $ is not a prime), although of course the fact that there is $ C ^ k $ Possibilities for $ c_i $ can be annoying, to say the least.

Of course, something Conditions are required $ a_ {i, j} $ For the system there are few solutions: If $ a_ {i, j} = 0 $ for all $ (i, j) $The system has many solutions!

ag.algebraic geometry – divisibility of a divisor

To let $ X $ to be a smooth complex projective curve and $ f colon X to Y $ an ├ętale 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 ├ętale 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 $?