Problems & Puzzles: Puzzles

 Puzzle 596. 12*n*(n+2)+1 JM Bergot* sent the following nice puzzle (See his A158463) 12*n*(n+2)+1 produces mainly primes or semiprimes. For example: from n=0 to 100 the only exceptions are n= 21, 38, 71, 76 & 96. Q1. Can you study this statistic a little more? Q2. Can you explain this statistic? ____ * As a matter of fact, Mr. Bergot sent the function 12*n*(n+2)+11, but I studied by mistake & typo in my Excel sheet 12*n*(n+2)+1. On top of that I missed the n=21 exception. Until my friends J. K. Andersen & Jean Brette let me know of it this morning... sorry, because on Saturday morning I'm changing several lines. The only justification in order to change the original function is because due to my typo I 'discovered' a better function than the original one for the production of primes & semiprimes...now I know what serendipity is!!!

Contributions came from J.K. Andersen, Luis Rodríguez, Paul Schmidt & Hakan Summakoğlu.

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Andersen wrote:

Euler's trinomial n^2+n+41 has no exceptions below 420.

It has to do with avoiding small factors. This will often give primes
or semiprimes for small numbers, but not for large numbers.

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Luis wrote:

n^2 + n + 247757 is a good polynomial too for this issue...

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Paul wrote:

This problem is related to the prime generating polynomials.  These polynomials are usually scored on the longest sequence of primes that are created.  Euler found the equation x^2 + x + 41.  This polynomial produces only primes or semiprimes in the range spedified.

Here is a list of some quadratics, some found on Wolfram (http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html) and the number of exceptions they produce where x = 0 thru 100:

 Equation Discoverer Exceptions X^2 + x + 41 Euler 0 X^2 – x + 41 Euler 0 36x^2 – 810x + 2753 Fung and Ruby 14 47x^2 – 1701x + 10181 Fung and Ruby 13 43x^2 – 537x + 2971 J. Brox 0 6x^2 – 342x + 4903 J. Brox 0 2x^2 + 29 Legendre 0 7x^2 – 371x + 4871 F. Gobbo 4 3x^2 + 39x + 37 A. Bruno 0 X^2 + x + 17 Legendre 0 4x^2 + 4x + 59 Honaker 0 2x^2 + 11 6 12x^2 + 24x + 1 Rivera 5 12x^2 + 24x + 11 Bergot 12

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Hakan wrote:

 range of n exceptions number of exceptions 1<=n<100 21,38,71,76,96 5 100<=n<200 109,112,131,132,145,153,157, 163,164,176,181,186,194 13 200<=n<300 208,214,219,225,241,249,250, 256,276,283,287,293,296,297,299 15 300<=n<400 300,307,310,317,318,334,337, 351,357,358,361,362,365,367,368, 371,373,377,378,380,384,386,395 23 400<=n<500 406,408,415,419,424,436,444,453, 463,472,473,480,481,482,487,492 16 500<=n<600 504,511,516,522,531,534,538, 540,546,548,560,563,569,572, 578,579,582,583,589,593,599 21 600<=n<700 604,614,618,626,630,633,641, 645,647,648,650,659,665,667, 681,686,691 17 700<=n<800 700,701,703,704,712,714,718, 725,727,734,742,743,747,752, 758,763,769,772,774,778,780, 783,793,795,796,799 26 800<=n<900 802,810,811,813,814,827,835, 846,847,848,850,852,861,868, 871,874,878,879,888,889 20 900<=n<1000 900,905,908,912,917,920,922,923, 929,946,951,952,956,961,963,966, 967,969,972,973,980,990,996,997 24

 range of n number of exceptions 1<=n<1000 180 1000<=n<2000 255 2000<=n<3000 279 3000<=n<4000 273 4000<=n<5000 309 5000<=n<6000 321 6000<=n<7000 304 7000<=n<8000 313 8000<=n<9000 333 9000<=n<10000 337

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