Problems & Puzzles: Puzzles

Puzzle 1169  A follow-up to Puzzle 782? Davide Rotondo, sent the following puzzle. Strange prime Producing Polynomial f(n) = n^2-76*n+1607 produces 78 successive prime integers, for n=0 to77 if for each odd n f(n) must be divided by 4. 1607,383,1459,347,1319,313,1187,281,1063,251,947,223,839,197,739,173,647,151,563,131,487,113,419,97,359, 83,307,71,263,61,227,53,199,47,179,43,167,41,163,41,167,43,179,47,199,53,227,61,263,71,307,83,359,97,419, 113,487,131,563,151,647,173,739,197,839,223,947,251,1063,281,1187,313,1319,347,1459,383,1607,421 Q. Can you produce another interesting  "strange prime producing polynomial"?

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From March 30 to April 5, cotribuion was made by J-M Rebert

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Rebert wrote:

f1(n)=n^2-79*n+1601 produces 80 successive prime integers, for n=0 to 79 if for each odd n f(n) must be divided by 2.

 [1601, 1523, 1447, 1373, 1301, 1231, 1163, 1097, 1033, 971, 911, 853, 797, 743, 691, 641, 593, 547, 503, 461, 421, 383, 347, 313, 281, 251, 223, 197, 173, 151, 131, 113, 97, 83, 71, 61, 53, 47, 43, 41, 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601]

 in which are 40 distinct primes:

 [41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033, 1097, 1163, 1231, 1301, 1373, 1447, 1523, 1601]

 

 f2(n)=abs(9n^2-408*n+4397) produces 43 successive prime integers, for n=0 to 42 if for each odd n f(n) must be divided by 2.

[4397, 1999, 3617, 1627, 2909, 1291, 2273, 991, 1709, 727, 1217, 499, 797, 307, 449, 151, 173, 31, 31, 53, 163, 101, 223, 113, 211, 89, 127, 29, 29, 67, 257, 199, 557, 367, 929, 571, 1373, 811, 1889, 1087, 2477, 1399, 3137]

in which are 41 distinct primes:

[29, 31, 53, 67, 89, 101, 113, 127, 151, 163, 173, 199, 211, 223, 257, 307, 367, 449, 499, 557, 571, 727, 797, 811, 929, 991, 1087, 1217, 1291, 1373, 1399, 1627, 1709, 1889, 1999, 2273, 2477, 2909, 3137, 3617, 4397]

 

f3(n)=abs(36n^2-738*n+1979) produces 44 successive prime integers, for n=0 to 43.

[1979, 1277, 647, 89, 397, 811, 1153, 1423, 1621, 1747, 1801, 1783, 1693, 1531, 1297, 991, 613, 163, 359, 953, 1619, 2357, 3167, 4049, 5003, 6029, 7127, 8297, 9539, 10853, 12239, 13697, 15227, 16829, 18503, 20249, 22067, 23957, 25919, 27953, 30059, 32237, 34487, 36809]

in which are 44 distinct primes:

[89, 163, 359, 397, 613, 647, 811, 953, 991, 1153, 1277, 1297, 1423, 1531, 1619, 1621, 1693, 1747, 1783, 1801, 1979, 2357, 3167, 4049, 5003, 6029, 7127, 8297, 9539, 10853, 12239, 13697, 15227, 16829, 18503, 20249, 22067, 23957, 25919, 27953, 30059, 32237, 34487, 36809]

 

f4(n)=abs(36n^2-2538n+36809) produces 45 successive prime integers, for n=0 to 44.

[36809, 34487, 32237, 30059, 27953, 25919, 23957, 22067, 20249, 18503, 16829, 15227, 13697, 12239, 10853, 9539, 8297, 7127, 6029, 5003, 4049, 3167, 2357, 1619, 953, 359, 163, 613, 991, 1297, 1531, 1693, 1783, 1801, 1747, 1621, 1423, 1153, 811, 397, 89, 647, 1277, 1979, 2753]

in which are 45 distinct primes:

[89, 163, 359, 397, 613, 647, 811, 953, 991, 1153, 1277, 1297, 1423, 1531, 1619, 1621, 1693, 1747, 1783, 1801, 1979, 2357, 2753, 3167, 4049, 5003, 6029, 7127, 8297, 9539, 10853, 12239, 13697, 15227, 16829, 18503, 20249, 22067, 23957, 25919, 27953, 30059, 32237, 34487, 36809]

...  I found by my own all the four polynomials but f1(n)=n^2-79*n+1601 & f4(n)=abs(36n^2-2538n+36809) were found also by others previously.

See:

https://mathworld.wolfram.com/Prime-GeneratingPolynomial.html

and

http://jeux-et-mathematiques.davalan.org/arit/pprime/index.html

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