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On Quadratic Polynomials Rich in Prime Numbers

Submitted:

02 March 2025

Posted:

03 March 2025

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Abstract
Prime numbers and methods of their generation have attracted mathematicians for centuries and, in the digital age, have found their applications in cryptography, signal processing and data compression, secure communications, hashing algorithms, cybersecurity, quantum computing algorithms, blockchain technology, and other areas. Prime numbers and prime generating polynomials were studied in [1-20]. There are many prime generating polynomials of different degrees \cite{weisstein2005} found so far; the most famous of them is \(x^2+x+41\) found by Euler in 1772 and \(x^2-x+41\) found by Legendre in 1798. Researchers are curious to find such a polynomial \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) that produces more prime numbers for given integer values of \(x\). This work is no exception, but the reader will not find any mathematical formulas or theorems as in mathematical works; instead, we want to show the result of our computational experiments in the programming language Julia, which in particular led to the discovery of quadratic polynomials that, similarly to Euler's prime generating polynomial, generate 40 primes. We also show that some of the currently known polynomials are not the richest in terms of the percentage of primes appearing in larger intervals, i.e. they produce fewer primes. For a better and more systematic understanding of what happens in prime number research, we demonstrate a video with network visualisation of keyword co-occurrence and co-authorship based on data from 7548 documents indexed in the Scopus database. Readers of this work are welcome to send me comments, suggestions, or proposals for collaborative research on prime numbers and their applications in science and engineering.
Keywords: 
prime numbers
;  
quadratic polynomial
;  
Julia programming language
Subject: 
Computer Science and Mathematics  -   Algebra and Number Theory

MSC:  Primary 11N05; Secondary 11N13; 11Y11

1. Where Are Prime Numbers Used?

Prime numbers and methods of generating them have fascinated mathematicians for centuries and, in the digital age, have found applications in cryptography, signal processing and data compression, secure communications, hashing algorithms, cybersecurity, quantum computing algorithms, blockchain technology, machinery and other areas. For example, in gear design, multiple gears often have a relatively prime number of teeth to ensure even wear of the teeth. For example, if two gears have 16 and 8 teeth, the same teeth of the first gear will always meet the same teeth of the second gear, and the wear of a single tooth can cause the wear of the tooth of the other gear.

2. Networks of Keywords and Co-Authors of Scopus Documents on Prime Numbers

The query TITLE-ABS-KEY ("prime number") was used in Scopus to find the related documents. Using the software VOSviewer1, we analysed the keywords of 7548 documents, and the visualisation of the keyword and co-authorship network is available at
https://youtu.be/L3zmETYtJGI.
Our analysis showed that the most popular keywords are cryptography, number theory, algorithms, public key cryptography, polynomials, finite fields, codes, data security, factorisation, matrix algebra, network security, RSA2, computational complexity, geometry, algebra, graph theory, computational theory, encryption, RSA algorithms, theorem proving, authentication, elliptic curve, and others.

3. Julia Code for Checking Prime Numbers

The computational experiments were carried out in the Julia3 v. 1.11.2 programming language on a computer with an Intel® CoreTM i5-10500 CPU at 3.10 GHz, 8.00 GB of RAM, 64-bit architecture, and Windows 11 operating system.
  • using Primes  # Import the Primes package
  • function compute_prime_stats()
  •     # Define the polynomial function
  •     f(x) = x^2 - x + 41
  •     total_count = 10^7
  •     prime_count = 0
  •     max_streak = 0
  •     current_streak = 0
  •     # Loop through values of x from 0 to total_count
  •     for x in 0:total_count
  •         value = f(x)
  •         if isprime(value)
  •             prime_count += 1
  •             current_streak += 1
  •         else
  •             max_streak = max(max_streak, current_streak)
  •             current_streak = 0
  •         end
  •         println("f($x) = $value -> Prime: ", isprime(value))
  •     end
  •     # Check in case the last numbers were primes
  •     max_streak = max(max_streak, current_streak)
  •     prime_percentage = (prime_count / total_count) * 100
  •     println("Percentage of prime numbers: $prime_percentage%")
  •     println("Maximum number of prime values: $max_streak")
  • end
  • # Execute the function
  • @time compute_prime_stats()
For x 2 − x + 41 prime generating polynomial for x m a x = 1 × 10 7 , in 27.230569 seconds (23 allocations: 1008 bytes), the program determined that 22.08% of the values were prime and recorded a maximum streak of 40 prime numbers.

4. Results of the Computational Experiment

In the following tables, a , b , c are the coefficients of a quadratic polynomial, and the other columns show how many primes are found in a given range.
Table 1. Prime-generating polynomials with positive a , b , c that produce 30 prime numbers (some of them can produce even more).
Table 1. Prime-generating polynomials with positive a , b , c that produce 30 prime numbers (some of them can produce even more).
No. a b c Number of primes Percentage
1 1 1 41 30 100.0
2 1 3 43 30 100.0
3 1 5 47 30 100.0
4 1 7 53 30 100.0
5 1 9 61 30 100.0
6 1 11 71 30 100.0
7 1 13 83 30 100.0
8 1 15 97 30 100.0
9 1 17 113 30 100.0
10 1 19 131 30 100.0
11 1 21 151 30 100.0
Table 2. Prime-generating polynomials that produce 40 prime numbers (note that some polynomials are likely to produce more primes and can be independently checked using the Julia code). Euler’s prime-generating polynomial and some others can also be found among the results.
Table 2. Prime-generating polynomials that produce 40 prime numbers (note that some polynomials are likely to produce more primes and can be independently checked using the Julia code). Euler’s prime-generating polynomial and some others can also be found among the results.
No. a b c Number of primes Percentage
1 1 -61 971 40 100.0
2 1 -59 911 40 100.0
3 1 -57 853 40 100.0
4 1 -55 797 40 100.0
5 1 -53 743 40 100.0
6 1 -51 691 40 100.0
7 1 -49 641 40 100.0
8 1 -47 593 40 100.0
9 1 -45 547 40 100.0
10 1 -43 503 40 100.0
11 1 -41 461 40 100.0
12 1 -39 421 40 100.0
13 1 -37 383 40 100.0
14 1 -35 347 40 100.0
15 1 -33 313 40 100.0
16 1 -31 281 40 100.0
17 1 -29 251 40 100.0
18 1 -27 223 40 100.0
19 1 -25 197 40 100.0
20 1 -23 173 40 100.0
21 1 -21 151 40 100.0
22 1 -19 131 40 100.0
23 1 -17 113 40 100.0
24 1 -15 97 40 100.0
25 1 -13 83 40 100.0
26 1 -11 71 40 100.0
27 1 -9 61 40 100.0
28 1 -7 53 40 100.0
29 1 -5 47 40 100.0
30 1 -3 43 40 100.0
31 1 -1 41 40 100.0
32 1 1 41 40 100.0
33 2 -88 997 40 100.0
34 2 -84 911 40 100.0
35 2 -80 829 40 100.0
36 2 -76 751 40 100.0
37 2 -72 677 40 100.0
38 2 -68 607 40 100.0
39 2 -64 541 40 100.0
40 2 -60 479 40 100.0
41 2 -56 421 40 100.0
42 2 -52 367 40 100.0
43 2 -48 317 40 100.0
44 2 -44 271 40 100.0
45 3 -105 941 40 100.0
46 6 -150 967 40 100.0
47 6 -138 823 40 100.0
48 6 -126 691 40 100.0
Table 3. One hundred prime generating polynomials with positive a , b , c and thirteen quadratic polynomials from [18] for values of x up to 10.000. Polynomials 9 and 10 from the bottom of the table can also be found in Table 3 (polynomials 56 and 21, respectively).
Table 3. One hundred prime generating polynomials with positive a , b , c and thirteen quadratic polynomials from [18] for values of x up to 10.000. Polynomials 9 and 10 from the bottom of the table can also be found in Table 3 (polynomials 56 and 21, respectively).
No. a b c Number of primes Percentage
1 2 44 43 4366 43.6556
2 2 40 1 4365 43.6456
3 1 1 41 4149 41.4859
4 1 3 43 4149 41.4859
5 1 5 47 4148 41.4759
6 6 6 31 3859 38.5861
7 6 18 43 3858 38.5761
8 4 2 41 3836 38.3562
9 4 10 47 3835 38.3462
10 2 40 19 3805 38.0462
11 4 6 43 3785 37.8462
12 1 35 23 3663 36.6263
13 9 9 43 3637 36.3664
14 9 3 41 3604 36.0364
15 9 15 47 3602 36.0164
16 24 12 31 3598 35.9764
17 2 34 31 3580 35.7964
18 24 36 43 3577 35.7664
19 16 4 41 3534 35.3365
20 16 20 47 3524 35.2365
21 3 39 37 3510 35.0965
22 3 33 1 3509 35.0865
23 7 49 41 3509 35.0865
24 10 20 29 3491 34.9065
25 22 22 17 3484 34.8365
26 2 4 31 3483 34.8265
27 2 8 37 3483 34.8265
28 2 12 47 3482 34.8165
29 16 12 43 3468 34.6765
30 25 25 47 3467 34.6665
31 1 23 23 3446 34.4566
32 1 25 47 3446 34.4566
33 1 21 1 3445 34.4466
34 25 5 41 3412 34.1166
35 7 7 17 3409 34.0866
36 7 21 31 3409 34.0866
37 1 37 29 3405 34.0466
38 38 40 1 3404 34.0366
39 33 15 11 3392 33.9166
40 36 18 43 3388 33.8766
41 36 30 47 3388 33.8766
42 35 35 19 3387 33.8666
43 36 6 41 3381 33.8066
44 25 15 43 3375 33.7466
45 43 3 1 3374 33.7366
46 49 35 47 3360 33.5966
47 47 5 1 3356 33.5566
48 41 1 1 3355 33.5466
49 32 18 47 3344 33.4367
50 20 20 43 3342 33.4167
51 49 21 43 3339 33.3867
52 49 7 41 3314 33.1367
53 34 46 23 3306 33.0567
54 1 27 13 3305 33.0467
55 1 29 41 3305 33.0467
56 3 3 23 3299 32.9867
57 3 9 29 3298 32.9767
58 3 15 41 3298 32.9767
59 34 22 11 3287 32.8667
60 40 40 29 3267 32.6667
61 8 8 31 3263 32.6267
62 8 24 47 3263 32.6267
63 8 16 37 3243 32.4268
64 4 46 23 3242 32.4168
65 23 35 1 3222 32.2168
66 14 6 29 3214 32.1368
67 14 34 49 3214 32.1368
68 2 30 29 3211 32.1068
69 2 26 1 3210 32.0968
70 26 32 43 3206 32.0568
71 46 46 17 3203 32.0268
72 28 42 31 3185 31.8468
73 33 33 41 3172 31.7168
74 28 14 17 3154 31.5368
75 14 22 37 3146 31.4569
76 18 12 31 3142 31.4169
77 18 36 47 3133 31.3269
78 42 30 29 3131 31.3069
79 22 14 11 3123 31.2269
80 15 45 1 3117 31.1669
81 26 20 37 3109 31.0869
82 4 42 1 3107 31.0669
83 4 50 47 3107 31.0669
84 15 15 17 3102 31.0169
85 15 45 47 3101 31.0069
86 7 49 23 3096 30.9569
87 18 24 37 3089 30.8869
88 13 27 1 3086 30.8569
89 46 20 47 3082 30.8169
90 11 9 11 3080 30.7969
91 11 31 31 3079 30.7869
92 11 13 13 3075 30.7469
93 11 35 37 3074 30.7369
94 25 17 17 3073 30.7269
95 33 15 19 3068 30.6769
96 32 32 37 3065 30.6469
97 1 19 17 3061 30.6069
98 22 30 19 3061 30.6069
99 1 21 37 3060 30.5969
100 26 26 37 3060 30.5969
1 36 -810 2753 3981 39.81
2 47 -1701 10181 3758 37.58
3 103 -4707 50383 3874 38.74
4 43 -537 2971 3808 38.08
5 8 -488 7243 4048 40.48
6 6 -342 4903 3874 38.74
7 2 0 29 3484 34.84
8 7 -371 4871 3526 35.26
9 3 3 23 3299 32.99
10 3 39 37 3510 35.10
11 1 1 17 2628 26.28
12 4 4 59 3408 34.08
13 2 0 11 2080 20.8

5. Conclusion

Using the Julia programming language, we have carried out a computational experiment and found quadratic polynomials rich in primes in a given range. Some of these polynomials may have already been found by others, so the reader can check them more carefully using the provided Julia code. The results can be updated and extended by increasing the range of x and a , b , c , which will, of course, increase the computational cost.

References

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1
https://www.vosviewer.com/
2
RSA (Rivest-Shamir-Adleman) is a public key cryptosystem, one of the oldest and most widely used for secure data transmission.
3
https://julialang.org/
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