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.

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2 Comments on Euler's (Prime to) Prime Generating Equation

  • Huen Yeong Kong -

    I believe I have a formula far better than Euler's formulae. It will generate consecutive primes from 2 to 7993. Here is the
    formula: factor(product(2*i,i,1,8000)); written in Maxima software. Try it. What do you think?

    Reply
  • Huen Yeong Kong -

    Here is a shortened version tor pagination problem:
    Numerical example:
    product(i,i,1,20);
    2432902008176640000
    factor(product(i,i,1,20));
    2^183^85^47^2111317*19
    cadd(m)::=[ppp:factor((m)^2),zzz:(makelist(part(ppp,k),k,1,length(ppp))),xxx:factor(zzz),yyy:makelist(first(part(xxx,j)),j,1,length(xxx))]$
    cadd(2432902008176640000);

    Output: ...........................,[2,3,5,7,11,13,17,19]]

    Reply

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