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Largest known prime number. As of January 2017, the largest known prime number is 274,207,281 â 1, a number with 22,338,618 digits. It was found in 2016 by the Great Internet Mersenne Prime Search (GIMPS). Plot of the number of digits in largest known prime by year, since the electronic computer. Note that the vertical scale is logarithmic. The red line is the exponential curve of best fit: y = exp(0.187394 t - 360.527), where t is in years.
There are many ways to find if a polynomial is prime. First, a prime polynomial is one that cannot be factored with real numbers. For example, the polynomial x^2-1 is not prime, because we can factor it to be (x+1)(x-1). However, the similar polynomial x^2+1 is in fact prime because we cannot factor it with the real numbers. I like to use some simple steps when determining whether or not a polynomial is prime. First, like fernando said, i look for simple things to factor out, like an x or 2.