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.