Prove whether this number is or is not prime [closed]

Question

Is the number 2438100000001 composite or prime?

Please first give a hint if you already know the answer.thanks!

Answer 1

Hint 1: Notice that $243$ and $81$ are powers of $3$, so the number has the form $$3^5\cdot10^{10}+3^4\cdot10^8+1$$

Hint 2: $x^5+x^4+1$ is reducible.



Hint 3: $x^5+x^4+1=(x^5+x^4+x^3)+(1-x^3)$.

Answer 2

[You already figured out the answer, so even though this is more than a hint, I hope I’m not giving anything away.]

If you notice the powers of $3$ hiding ($243$ and $81$), you may think to rewrite this big number as $$2438100000001=3^5\cdot10^{10}+3^4\cdot10^8+1.$$ Then you might write $$2438100000001=300^5+300^4+1.$$ You might not see how to factor this, but you can look for a pattern for factoring similar expressions with smaller numbers than $300$.

$$\begin{align} &1^5+1^4+1& =\quad&3& =\quad&\color{blue}3\cdot 1\\ &2^5+2^4+1& =\quad&49& =\quad&\color{blue}7\cdot 7\\ &3^5+3^4+1& =\quad&325& =\quad&\color{blue}{13}\cdot 25\\ &4^5+4^4+1& =\quad&1281& =\quad&\color{blue}{21}\cdot 61\\ &5^5+5^4+1& =\quad&3751& =\quad&\color{blue}{31}\cdot 341\\ &\cdots\\ &300^5+300^4+1& =\quad& 2438100000001& =\quad& ?\cdot ?\\ \end{align}$$

Notice that if you happen to factor these numbers the way I show, the differences between the first factors form an arithmetic sequence. The first factors are $\color{blue}3, \color{blue}3+\color{red}4=\color{blue}7, \color{blue}7+\color{red}6=\color{blue}{13}, \color{blue}{13}+\color{red}8=\color{blue}{21}, \color{blue}{21}+\color{red}{10}=\color{blue}{31},\dots$ (You have to play with the factorizations to see this — the factor I wrote first isn’t always the smallest prime factor, or even a prime, so you have to play around.)

When a sequence of numbers has differences that are an arithmetic sequence, there’s a formula for that sequence using multiples of $n^2$, $n$, and constants. If you knew that, you’d notice that the sequence $(3,7,13,21,31,\dots)$ is $(1^2+2,2^2+3,3^2+4,4^2+5, 5^2+6,\dots)$, and you might guess that $90301=300^2+301$ is a factor of your original number.

And you would be right.

In somewhat less time, you could try prime factors and find $73$, but finding the factor $90301$ (which is $73\cdot1237$) by hand is more interesting!

Answer 3

To answer the question: $2438100000001$ is a composite number.

$2438100000001 = 73 \dot\ 829 \dot\ 1237 \dot \ 32569$

Answer 4

Hint: Multiply 1025473 by 2377537. Neither of these two numbers is prime, but you should easily be able to factor them with the help of an ordinary scientific calculator and a list of small primes.

EDIT: You said

Please first give a hint if you already know the answer.

That's exactly what I'm doing by showing you that 2438100000001 is divisible by numbers other than 1 and itself. I already know the answer because this number, as forbidding as it may look, it's quite small compared to the numbers computers are routinely factoring these days. If you don't have Mathematica, try putting this question to Wolfram Alpha: "factor 2438100000001." With a decent Internet connection, it should give you the answer after only a very short wait.