# Euler’s (Prime to) Prime Generating Equation

The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent but still stun the mathematicians. The prime generating equation by Euler is a very specific binomial equation on prime numbers and yields more primes than any other relations out there in number theory. Euler told that the equation $f(x)=x^2+x+k$ yields many prime numbers with the values of x being input from x=0 to $x=k-2$ ; k being a prime.

Let’s see how many primes we can get by using different values of $k$ and $x$:

 Serial Number Value of k (prime) Value of x (from x=0 to x=k-2) Value of f(x)=(x^2+x)+k Not a Prime? 1 2 0 2 2 3 0 3 3 1 5 4 5 0 5 5 1 7 6 2 11 7 3 17 8 7 0 7 9 1 9 No 10 2 13 11 3 19 12 4 27 No 13 5 37 14 11 0 11 15 1 13 16 2 17 17 3 23 18 4 31 19 5 41 20 6 53 21 7 67 22 8 83 23 9 101 24 13 0 13 25 1 15 No 26 2 19 27 3 25 No 28 4 33 No 29 5 43 30 6 55 No 31 7 69 No 32 8 85 NO 33 9 103 34 10 123 No 35 11 145 No 36 17 0 17 37 1 19 38 2 23 39 3 29 40 4 37 41 5 47 42 6 59 43 7 73 44 8 89 45 9 107 46 10 127 47 11 149 48 12 173 49 13 199 50 14 227 51 15 257 52 19 0 19 53 1 21 54 2 25 No 55 3 31 56 4 39 No 57 5 49 58 6 61 59 7 75 No 60 8 91 No 61 9 109 62 10 129 No 63 11 151 64 12 175 No 65 13 201 No 66 14 229 67 15 259 68 16 291 69 17 325 No 70 23 0 23 71 1 25 No 72 2 29 73 3 35 No 74 4 43 75 5 53 76 6 65 No 77 7 79 78 8 95 No 79 9 113 80 10. 133 81 11 155 No 82 12 179 83 13 205 No 84 14 233 85 15 263 86 16 295 No 87 17 329 88 18 365 No 89 19 403 90 20 443 91 21 485 No

The above table yields many prime numbers, which again can be put at the place of k and so on the table can be progressed.

According to Euler, 41 was the most appropriate value of k yielding more prime numbers than any other k. In the list below, each value of f(x) is a prime for k=41:

 k= 41 x= 0 f(x)= 41 1 43 2 47 3 53 4 61 5 71 6 83 7 97 8 113 9 131 10 151 11 173 12 197 13 223 14 251 15 281 16 313 17 347 18 383 19 421 20 461 21 503 22 547 23 593 24 641 25 691 26 743 27 797 28 853 29 911 30 971 31 1033 32 1097 33 1163 34 1231 35 1301 36 1373 37 1447 38 1523 39 1601

So, the Euler’s Prime Generating Equation can be written as
$f(x) = x^2+x+41$ ; where $x$ is an integer ranging from 0 to 39.

What if we increase the value of x beyond the limit of 39? What will we get?

The next values of f(x) in this series would be 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393, … .
Are all these prime numbers too? The answer is no. 1681 is not a prime number, neither are 1763 and 2021. Though all others (in above sequence at-least) are prime numbers.