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 NumberValue of k (prime)Value of x (from x=0 to x=k-2)Value of f(x)=(x^2+x)+kNot a Prime?
1202
2303
315
4505
517
6211
7317
8707
919No
10213
11319
12427No
13537
1411011
15113
16217
17323
18431
19541
20653
21767
22883
239101
2413013
25115No
26219
27325No
28433No
29543
30655No
31769No
32885NO
339103
3410123No
3511145No
3617017
37119
38223
39329
40437
41547
42659
43773
44889
459107
4610127
4711149
4812173
4913199
5014227
5115257
5219019
53121
54225No
55331
56439No
57549
58661
59775No
60891No
619109
6210129No
6311151
6412175No
6513201No
6614229
6715259
6816291
6917325No
7023023
71125No
72229
73335No
74443
75553
76665No
77779
78895No
799113
8010.133
8111155No
8212179
8313205No
8414233
8515263
8616295No
8717329
8818365No
8919403
9020443
9121485No

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
143
247
353
461
571
683
797
8113
9131
10151
11173
12197
13223
14251
15281
16313
17347
18383
19421
20461
21503
22547
23593
24641
25691
26743
27797
28853
29911
30971
311033
321097
331163
341231
351301
361373
371447
381523
391601

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.

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