<s>[INST] Factor 90301 without the aid of the computer Computer break down easily know $90301=73\cdot1237$
Is there any way I want, 
without the aid of the computer to determine 90301
 is a prime number or Composite number
 [/INST] With a little insight and with a little luck we can find the smallest prime factor quite easily.
Suppose prime $\rm\:p\ |\ 90301 = a^2+a+1 = (a^3-1)/(a-1),\:$ for $\rm\:a = 300.\:$ Therefore, $\rm\:mod\ p\!:\ a^3\equiv 1,\:$  so if $\rm\:a\not\equiv 1\:$ then $\rm\:a\:$ has order $3,$ so $\rm\:3\ |\ p-1.\:$ But $\rm\:a\equiv 1\:\Rightarrow\:p\ |\ 3\nmid 90301.$
So every prime divisor of $90301$ is $\equiv 1\pmod 3.\:$ We might  deduce stronger divisor congruences if $\rm\:a\:$ is a power, e.g. $\rm\:a \equiv c^2\:$ $\Rightarrow$ $\rm\:c^{6}\equiv 1,\:$ so if $\rm\:c^3\not\equiv 1\:$ then $\rm\:c\:$ has order $6,\:$ so $\rm\:6\ |\ p-1.\:$ Similarly  $\rm\:a\equiv c^3\:$  $\Rightarrow$ $\rm\:c^{9}\equiv 1,\:$ so if $\rm\:c^3\not\equiv 1\:$ then $\rm\:c\:$ has order $9,\:$ so $\rm\:9\ |\ p-1.\:$ 
The first case $\rm\:a\equiv c^2\:$ doesn't yield easy solutions but the second case $\rm\:a\equiv c^3\:$ strikes gold  quickly. We seek primes $\rm p$ such that $\rm\: a = 300 \equiv c^3,\:$ i.e. $\rm\:p\ |\ 300- c^3.\:$ For $\rm\:c=1\!:\ 300-1^2 = 13\cdot 23\:$ but neither factor is $\equiv 1\pmod 9.\:$ For $\rm\:c=2\!:\ 300-2^3 = 4\cdot 73\:$ and $\:73\equiv 1\pmod 9$ and this works: $\: 73\ |\ 90301.$
Note $\ $ Often properties of numbers are specializations of those of functions (e.g. polynomials), as above where integer factorizations arise by specializing factorizations of cyclotomic polynomials. Here are some other useful examples. Aurifeuille, Le Lasseur and Lucas discovered the so-called Aurifeuillian factorizations of cyclotomic polynomials $\rm\;\Phi_n(x) = C_n(x)^2 - n\ x\  D_n(x)^2\;$. These play a role in factoring numbers of the form $\rm\; b^n \pm 1\:$, cf. the Cunningham Project. Below are some simple examples of such factorizations:
$$\begin{array}{rl}
x^4 + 2^2 \quad=&  (x^2 + 2x + 2)\;(x^2 - 2x + 2) \\
\frac{x^6 + 3^2}{x^2 + 3} \quad=&  (x^2 + 3x + 3)\;(x^2 - 3x + 3) \\\\
\frac{x^{10} - 5^5}{x^2 - 5} \quad=&  (x^4 + 5x^3 + 15x^2 + 25x + 25)\;(x^4 - 5x^3 + 15x^2 - 25x + 25) \\\\
\frac{x^{12} + 6^6}{x^4 + 36} \quad=&  (x^4 + 6x^3 + 18x^2 + 36x + 36)\;(x^4 - 6x^3 + 18x^2 - 36x + 36) \\\\
\end{array}$$
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