Primes in arithmetic progression: Difference between revisions

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This also shows that an AP with common difference <math>a</math> cannot contain more consecutive prime terms than the value of the smallest prime that does not divide <math>a</math>.

If <math>k</math> is prime then an AP-<math>k</math> can begin with <math>k</math> and have a common difference which is only a multiple of <math>(k-1)\#</math> instead of <math>k\#</math>. (From H. J. Weber, ``Less Regular Exceptional and Repeating Prime Number Multiplets," arXiv:1105.4092[math.NT], Sect.3.) For example, the AP-3 with primes <math>\{3,5,7\}</math> and common difference <math>2\#=2</math>, or the AP-5 with primes <math>\{5,11,17,23,29\}</math> and common difference <math>4\#-=6</math>. It is conjectured that such examples exist for all primes <math>k</math>. {{As of|2018}}, the largest prime for which this is confirmed is <math>k=19</math>, for this AP-19 found by Wojciech Iżykowski in 2013:

:<math>19 + 4244193265542951705\cdot 17\#\cdot n</math>, for <math>n=0</math> to <math>18</math>.<ref name="APrecords" />

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{| class="wikitable"

|+ Largest known AP-''k'' {{as of|2025|45|lc=o1}}<ref name="APrecords">Jens Kruse Andersen and Norman Luhn, [https://www.pzktupel.de/JensKruseAndersen/aprecords.htm''Primes in Arithmetic Progression Records'']. Retrieved 2023-12-11.</ref>

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! ''k'' !! Primes for ''n'' = 0 to ''k''&minus;1 !! Digits !! Year !! Discoverer

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| (65502205462 13088317669 + 63172808286383832302·''n'')·23712399# + 1 ||align="right" | 10141034 || 20122025 || Ken Davis, PaulNorman UnderwoodLuhn

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! 10