### number theory - Minimal positive integer n which is divisible by d and has sum of digits equal to s

I found this problem on codeforces (http://codeforces.com/problemset/problem/1070/A) and I'm trying to understand a pretty elegant solution that was posted:#include<bits/stdc++.h>using namespace std;typedef long long ll;int d,s;struct node{ int mod,sum; char s[700]; int len;};queue<node> Q;bool v[512][5200];int main(){ scanf("%d%d",&d,&s); Q.push({0,0,0,0}); while(!Q.empty()) { node a=Q.front();Q.pop(); for(int i=0;i<10;i++) { node b=a; b.s[b.len++]=i+'0'; ...Read more

### error correction - Code that maps numbers from one number to another with where each number has a distance greater than 1

I need to tag a load of books with a unique id. Because human error would really mess with the system i need the code to detect if one of the numbers is wrong. That means that no two elements of the code can have a hamming distance of 1. Or have a parity check method or something again such that some errors can be detected. I would normally post what I've done so far, but I don't know where to start really.Thanks...Read more

### number theory - Modular arithmetic. How to solve the following equation?

How to solve the following equation?I am interested in the methods of solutions.n^3 mod P = (n+1)^3 mod PP- Prime numberShort example with the answer. Could you gives step-by-step solutions for my example.n^3 mod 61 = (n + 1)^3 mod 61Integer solutions:n = 61 m + 4, n = 61 m + 56, m element ZZ - is set of integers....Read more

### number theory - Calculating sum of geometric series (mod m)

I have a series S = i^(m) + i^(2m) + ............... + i^(km) (mod m) 0 <= i < m, k may be very large (up to 100,000,000), m <= 300000I want to find the sum. I cannot apply the Geometric Progression (GP) formula because then result will have denominator and then I will have to find modular inverse which may not exist (if the denominator and m are not coprime).So I made an alternate algorithm making an assumption that these powers will make a cycle of length much smaller than k (because it is a modular equation and so I would obtai...Read more

### Number Theory Congruence Modulo

I’m self-studying number theory using George Andrew’s textbook.I’m at the chapter of congruence modulo. There is one or two parts that I couldn’t quite figure out. Wonder if someone could point things out for me.By definition, if c≠0, a≡b(mod c) provided that (a-b)/c is an integer. That is c|(a-b).If a= 5, b=-3, c=85 is congruent to -3 modulo 8, 5≡-3(mod 8) since (5-(-3))/8 is an integer of 1.I read else where that congruence modulo could also be interpreted as the remainder of (a/c) is equal to the remainder of (b/c).If that’s the case, using ...Read more

### How to find total number of divisors upto N?

Given a number N, have to find number the divisors for all i where i>=1 and i<=N. Can't figure it out.Do I have to this using prime factorization? Limit is N<=10^9Sample Output:1 --> 12 --> 33 --> 54 --> 85 --> 106 --> 147 --> 168 --> 209 --> 2310 --> 2711 --> 2912 --> 3513 --> 3714 --> 4115 --> 45...Read more

### number theory - Let $p$ be prime and $0<a< p$ an integer such that $\Big({a\over p}\Big) =1$ Then there exists integers $x,y$ such that $p=x^2-ay^2$.

Let $p$ be prime and $0<a< p$ an integer such that $$\Big({a\over p}\Big) =1$$Then there exists integers $x,y$ such that $p=x^2-ay^2$.Edit: For which primes is this true? From CalebKoch answer we see this is not true in general. Can you sugest me a literature where I can find a theory of this.I suspect, if it is true, then it has something to do with a Thue theorem. http://mathworld.wolfram.com/ThuesTheorem.htmlA found this interesting results:https://en.m.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares...Read more

### number theory - Bound for divisor function

I have been searching for a bound of the divisor function $d(n)$, meaning the number of divisors of n. So far I have found that it can be bounded by $$d(n) \le e^{O(\frac{\log n}{\log \log n})}$$Wigert has proven the constant is $\log 2$ so $$d(n) \le e^{(\log 2+ o(1)) \frac{\log n}{\log \log n}}$$However, when I tried to check that bound on a computer, it did not seem right. I have drawn $d(n)$ (in blue), $e^{\frac{\log n}{\log \log n}}$ (in red) and $e^{\log 2 \frac{\log n}{\log \log n}}$ (in green) on the following graph: Furthermore, wh...Read more

### number theory - Can one show a beginning student how to use the $p$-adics to solve a problem?

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the $p$-adics are useful.As a graduate student in algebraic number theory, this question is easy to answer. But I'm wondering if there's a way to answer this question to someone who only knows very basic things about number theory and the $p$-adics. I'm thinking of someone who has learned over the course of a few days what the $p$-adic numbers are, what t...Read more

### number theory - Calculating the median in the St. Petersburg paradox

I am studying a recreational probability problem (which from the comments here I discovered it has a name and long history). One way to address the paradox created by the problem is to study the median value instead of the expected value. I want to calculate the median value exactly (not only find bounds or asymptotic values). I have found a certain approach and I am stuck in a specific step. I present my analysis and I would like some help on that specific step. [Note: Other solutions to the general problem are welcome (however after the revel...Read more

### number theory - IMO 1997 problem 6

For each positive integer $n$ , let $f (n)$ denote the number of ways of representing $n$ as a sum of powers of $2$ with non-negative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. Prove that, for any integer $n \geqslant 3$, $$2^{n^2/4} < f (2^n) < 2^{n^2/2}.$$...Read more

### number theory - Prove that $lcm(a , b) = \prod_{i=1} (P_i)^{\max(\alpha_i,\beta_i)}$

we are given that $a = p_1^{\alpha 1} .... p_k^{\alpha k}$ and $b = p_1^{\beta 1} .... p_k^{\beta k}$. Where $p_1 ... p_k$ are pairwise distinct primes and $\alpha_i$ and $\beta_i$ are non negative integers.I tried expanding out on the left hand side using the formula for the LCM but i couldn't see where to go from there....Read more