Wednesday, 30 July 2025

A Well-Known Legendre Prime-Producing Polynomial

 Here is what Gemini had to say about the prime-producing polynomial \( 2 k^2+29\):

The polynomial \( 2 k^2+29\) is a well-known example of a prime-producing polynomial, often associated with the French mathematician Adrien-Marie Legendre. Here's a breakdown of its discovery, usage, and prime-producing potential:

Discovery:

  • Adrien-Marie Legendre (1752-1833) is credited with the discovery of the polynomial \( 2  k^2+29\). He published this result in his work "Essai sur la théorie des nombres" in 1798.

  • This came in the wake of significant work on prime-generating polynomials, notably Euler's polynomial \(k^2+k+41\), which generates primes for 40 consecutive integer values. Legendre's work further explored this fascinating property in different polynomial forms.

Usage and Significance:

  • Illustrates Prime-Generating Properties: While no non-constant polynomial with integer coefficients can generate only prime numbers for all integer values (this has been proven), polynomials like \( 2 k^2+29\) are significant because they produce a surprisingly long sequence of prime numbers for consecutive integer inputs.

  • Connection to Class Number Theory: The behavior of such prime-producing quadratic polynomials is deeply connected to the theory of class numbers in imaginary quadratic fields. The fact that \( 2 k^2+29\) (and other similar polynomials like Euler's \(k^2+k+41\)) produce so many primes is not a coincidence but is tied to specific properties of the corresponding quadratic fields.

  • Educational and Recreational Mathematics: These polynomials are frequently used in number theory courses and popular science to demonstrate the elusive nature of prime number distribution and the intriguing patterns that can emerge. They serve as excellent examples to explore the question of "prime-generating formulas."

  • In Popular Culture: The sequence of primes generated by \( 2 k^2+29\) has even appeared in popular culture, such as in Ken Follett's novel "Code to Zero."

Prime-Producing Potential:

  • The polynomial \( \text{P}(k)= 2 k^2+29 \) generates prime numbers for a remarkable number of consecutive integer values of \(k\). Specifically, it produces primes for \(k=0,1,2, \dots 28\). This means it generates 29 consecutive primes starting from .

  • Let's list the first few:

    • \( \text{P}(0)=2 \times 0^2+29=29 \) (prime)

    • \( \text{P}(1)=2 \times 1^2+29=31\) (prime)

    • \( \text{P}(2)=2 \times 2^2+29=37 \) (prime)

    • \( \text{P}(3)=2 \times 3^2+29=47 \) (prime)

    • ...and so on, up to \( \text{P}(28) \).

  • For \(k=29\)$$\text{P}(29)=2 \times 29^2+29=29 \, (2×29+1)=29 \times 59$$which is a composite number. This illustrates the general principle that no such polynomial can produce primes indefinitely.

In summary, Legendre's polynomial \( 2 k^2+29\) is a notable discovery in number theory, demonstrating an impressive string of prime outputs for consecutive integer inputs. Its significance lies in its historical context alongside other prime-generating polynomials like Euler's, its connection to deeper number theoretic concepts like class numbers, and its role as a compelling example in the study of prime numbers.

The polynomial popped up because of a reference to it in OEIS A241554: composite numbers of the form \(2  k^2 + 29\). The number associated with my diurnal age today, 27877, is a member of this sequence:$$ \begin{align} \text{If }k &=118 \\ 2k^2+29 &= 2 \times 118^2+29 \\ &= 27877 \\ &=61 \times 457 \end{align}$$When this polynomial is not busy churning out primes, it is busy churning out semiprimes. In fact, it is only when \(k=185\) that we get a number that is not a prime or a semiprime:$$\begin{align} \text{If }k &=185 \\ 2k^2+29 &= 2 \times 185^2+29 \\ &= 68479 \\ &=31 \times 47^2 \end{align}$$Follow this permalink to generate a table of these numbers and their factors. Here is a list of the members of OEIS A241554 between 1 and 40000:

1711, 1829, 2077, 2479, 3071, 3901, 5029, 6527, 6757, 7471, 7967, 8479, 10397, 10981, 11581, 14141, 15167, 15517, 15871, 16591, 16957, 17701, 18079, 18847, 19631, 20837, 22927, 23791, 25567, 26941, 27877, 28829, 29797, 30287, 31279, 31781, 32287, 35941, 38117

The table below shows the factorisations of the previous numbers as well as the corresponding \(k\) values:

  k     2*k^2+29   factors

  29    1711       29 * 59
  30    1829       31 * 59
  32    2077       31 * 67
  35    2479       37 * 67
  39    3071       37 * 83
  44    3901       47 * 83
  50    5029       47 * 107
  57    6527       61 * 107
  58    6757       29 * 233
  61    7471       31 * 241
  63    7967       31 * 257
  65    8479       61 * 139
  72    10397      37 * 281
  74    10981      79 * 139
  76    11581      37 * 313
  84    14141      79 * 179
  87    15167      29 * 523
  88    15517      59 * 263
  89    15871      59 * 269
  91    16591      47 * 353
  92    16957      31 * 547
  94    17701      31 * 571
  95    18079      101 * 179
  97    18847      47 * 401
  99    19631      67 * 293
  102   20837      67 * 311
  107   22927      101 * 227
  109   23791      37 * 643
  113   25567      37 * 691
  116   26941      29 * 929
  118   27877      61 * 457
  120   28829      127 * 227
  122   29797      83 * 359
  123   30287      31 * 977
  125   31279      31 * 1009
  126   31781      61 * 521
  127   32287      83 * 389
  134   35941      127 * 283
  138   38117      47 * 811

There are 103 primes in the range up to 40000 that are generated by this polynomial. These start with \(k=0\):

29, 31, 37, 47, 61, 79, 101, 127, 157, 191, 229, 271, 317, 367, 421, 479, 541, 607, 677, 751, 829, 911, 997, 1087, 1181, 1279, 1381, 1487, 1597, 1951, 2207, 2341, 2621, 2767, 2917, 3229, 3391, 3557, 3727, 4079, 4261, 4447, 4637, 4831, 5231, 5437, 5647, 5861, 6079, 6301, 6991, 7229, 7717, 8221, 8741, 9007, 9277, 9551, 9829, 10111, 10687, 11279, 11887, 12197, 12511, 12829, 13151, 13477, 13807, 14479, 14821, 16229, 17327, 18461, 19237, 20029, 20431, 21247, 21661, 22079, 22501, 23357, 24229, 24671, 25117, 26021, 26479, 27407, 28351, 29311, 30781, 32797, 33311, 33829, 34351, 34877, 35407, 36479, 37021, 37567, 38671, 39229, 39791

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