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!!!

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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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