Mathematics

Questions tagged [congruences]

For questions about congruences in modular arithmetic, concerning for example the chinese remainder theorem, Fermat's little theorem or Euler's totient theorem.

1702 questions
90
votes
5 answers

How to solve these two simultaneous "divisibilities" : $n+1\mid m^2+1$ and $m+1\mid n^2+1$

Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. Thanks !
user14479
81
votes
3 answers

Mathematicians shocked(?) to find pattern in prime numbers

There is an interesting recent article " Mathematicians shocked to find pattern in "random" prime numbers " in New Scientist . (Don't you love math titles in the popular press? Compare to the source paper's Unexpected Biases in the Distribution of…
Tito Piezas III
  • 61,356
48
votes
8 answers

Why is $a^n - b^n$ divisible by $a-b$?

I did some mathematical induction problems on divisibility $9^n$ $-$ $2^n$ is divisible by 7. $4^n$ $-$ $1$ is divisible by 3. $9^n$ $-$ $4^n$ is divisible by 5. Can these be generalized as $a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer? But why…
z_z
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30
votes
13 answers

Why do we use "congruent to" instead of equal to?

I'm more familiar with the notation $a \equiv b \pmod c$, but I think this is equivalent to $a \bmod c = b \bmod c $, which makes it clear that we should put a $=$ instead of $\equiv$. What's the reason for the change of sign? If it's to emphasize…
YoTengoUnLCD
  • 13,752
20
votes
4 answers

Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?

I know that if the number is a perfect square then it will be congruent to $0$ or $1$ (mod $4$). Now since the number is even, I know that it is either $0$ or $2$ (mod $4$). How would I go about answering this?
user5826
  • 12,634
19
votes
2 answers

Solve congruence: $45x \equiv 15 \pmod{78}$ (What am I doing wrong?)

Question about solving congruence. I've worked out how to solve them for the most part except for the following problem I'm having: $$45x \equiv 15 \pmod{78}$$ By the euclidean algorithm, I work out that the gcd of 45 and 78 = 3 which means there…
Arvin
  • 1,761
18
votes
1 answer

Flaw or no flaw in MS Excel's RNG?

I have a question about my understanding of an article of B.D. McCullough (2008) about Excel's implementation of the Wichmann-Hill random number generator (1982). First, a bit of context The Wichmann-Hill algorithm is given in AS 183 here . As one…
Jean-Claude Arbaut
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15
votes
2 answers

What is the difference between Hensel lifting and the Newton-Raphson method?

So in the Newton-Raphson method to iteratively approximate a root of a real polynomial, we start with a crude approximation $x_0 \in \mathbb{R}$ for $f(x)=0$ where $f(x) \in \mathbb{R}[x]$. For the next iterate $x_1$, we put $x_1 = x_0 + \epsilon$,…
BharatRam
  • 2,637
14
votes
3 answers

Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$.

So the problem states: Show that for every prime $p$, there is an integer $n$ such that $2^{n}+3^{n}+6^{n}-1$ is divisible by $p$. I was thinking about trying to prove this using the corollary to Fermat's Little Theorem, that for every prime $p$,…
mataxu
  • 195
13
votes
4 answers

Divisibility rules and congruences

Sorry if the question is old but I wasn't able to figure out the answer yet. I know that there are a lot of divisibility rules, ie: sum of digits, alternate plus and minus digits, etc... but how can someone derive those rules for any number $n$…
Tarantula
  • 216
12
votes
6 answers

Proof that there are infinitely many primes congruent to 3 modulo 4

I'm having difficult proving this. As a hint the exercise to prove first, that if $a\lneqq \pm 1$ satisfies $a \equiv 3 \pmod4$, then exist $p$ prime, $p \equiv 3 \pmod 4$ such $p\mid4$. But I'm not really getting for what purpose can this be used.
FranckN
  • 1,354
12
votes
1 answer

When is $(p-2)!-1$ power of $p$ if $p$ is prime?

If $p$ is prime, for what values of $p$ is $(p-2)!-1$ a power of $p$? I know how to solve that when $p<5$ then $(p-1)!+1$ can be written as power of $p$.
Yogesh Ghaturle
  • 335
12
votes
4 answers

Is there a sequence of 5 consecutive positive integers such that none are square free?

Is there a sequence of 5 consecutive positive integers such that none are square free? A number is square free if there is no prime number p such that $p^2 \mid n$ What I've tried doing so far is to show that for every sequence of 5 integers there…
andrew749
  • 215
11
votes
2 answers

Identity involving pentagonal numbers

Let $G_n = \tfrac{1}{2}n(3n-1)$ be the pentagonal number for all $n\in \mathbb{Z}$ and $p(n)$ be the partition function . I was trying to prove one of the Ramanujan's congruences : $$p(5n-1) = 0 \pmod 5,$$ and my "brute force" proof reduces to show…
Zilin J.
  • 4,310
11
votes
2 answers

Show $a^p \equiv b^p \mod p^2$

I am looking for a hint on this problem: Suppose $a,b\in\mathbb{N}$ such that $\gcd\{ab,p\}=1$ for a prime $p$. Show that if $a^p\equiv b^p \pmod p$, then we have: $$a^p \equiv b^p \pmod {p^2}.$$ I have noted that $a,b$ are necessarily coprime to…
anak
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