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 2022-02-27, 16:57 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 5×233 Posts a puzzle from 'mathematics magazine' As a lifetime member of the Mathematical Association of America, I get their magazines. It is fun to read the articles, and challenge myself with the puzzles. From the December 2021 issue (Volume 94, Number 5) I read that (in the 'quickies' section), problem number 1116, Proposed by Lokman Gokce of Turkey, Problem - For which prime numbers p is A = p^4-35*p^3+365*p^2-1225*p+1259 a prime number? I plugged in 2,3,5, and 11, and those are solutions. Maple helped me find a short list of prime solutions. {2, 3, 5, 11, 19, 25, 45, 55, 65, 75, 105} OEIS.org does not recognize this sequence. Here is my code and output. Code: > for b from 3 by 2 to 301 do c := b^4-35*b^3+365*b^2-1225*b+1259; if isprime(c) then print("a solution is ", b, c) end if end do; 5 "a solution is ", 3, 5 509 "a solution is ", 5, 509 965 845 5 "a solution is ", 11, 5 -1315 -2491 -2515 5 "a solution is ", 19, 5 6845 20165 42509 "a solution is ", 25, 42509 76805 126365 194885 286445 405509 556925 745925 978125 1259525 1596509 "a solution is ", 45, 1596509 1995845 2464685 3010565 3641405 4365509 "a solution is ", 55, 4365509 5191565 6128645 7186205 8374085 9702509 "a solution is ", 65, 9702509 11182085 12823805 14639045 16639565 18837509 "a solution is ", 75, 18837509 21245405 23876165 26743085 29859845 33240509 36899525 40851725 45112325 49696925 54621509 59902445 65556485 71600765 78052805 84930509 "a solution is ", 105, 84930509 92252165 100036445 108302405 117069485 126357509 "a solution is ", 115, 126357509 136186685 146577605 157551245 169128965 181332509 194184005 207705965 221921285 236853245 252525509 "a solution is ", 135, 252525509 Does anyone know the answer to this? I will probrably have to wait two years for the magazine to publish the result. Have a nice day. Matt
2022-02-27, 19:51   #2
Dr Sardonicus

Feb 2017
Nowhere

589810 Posts

(my emphasis)
Quote:
 Originally Posted by MattcAnderson As a lifetime member of the Mathematical Association of America, I get their magazines. It is fun to read the articles, and challenge myself with the puzzles. From the December 2021 issue (Volume 94, Number 5) I read that (in the 'quickies' section), problem number 1116, Proposed by Lokman Gokce of Turkey, Problem - For which prime numbers p is A = p^4-35*p^3+365*p^2-1225*p+1259 a prime number? I plugged in 2,3,5, and 11, and those are solutions. Maple helped me find a short list of prime solutions. {2, 3, 5, 11, 19, 25, 45, 55, 65, 75, 105}
Your reading skills leave much to be desired.

Last I checked, 25, 45, 55, 65, 75, and 105 weren't prime numbers.

 2022-02-28, 02:21 #3 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 5×233 Posts probably the right answer oops You are right, those 'p' values are composite numbers. At least I tried a hard problem. here is my second try. "If at first, you don't succeed, try, try again." Code: > factor(x^2-1); (x - 1) (x + 1) > factor(p^4-35*p^3+365*p^2-1225*p+1259); 4 3 2 p - 35 p + 365 p - 1225 p + 1259 > ifactor(1225); 2 2 (5) (7) > ifactor(1259); (1259) > c := p^4-35*p^3+365*p^2-1225*p+1259; 4 3 2 p - 35 p + 365 p - 1225 p + 1259 > p := 19; 19 > c; 6520646525 > ifactor(c); (5) (1259) (144983) (2677) > p := 2; 2 > c; 5 > > for b from 2 to 10000 do if isprime(b) then c := b^4-35*b^3+365*b^2-1225*b+1259; if isprime(c) then print("a solution is ", b, c) end if end if end do; "a solution is ", 2, 5 "a solution is ", 3, 5 "a solution is ", 5, 509 "a solution is ", 11, 5 "a solution is ", 19, 5 My claim is that there are five prime solutions for p. They are 2,3,5,11, and 19. I searched p to 10,000. I would like confirmation from someone else. Also, close inspection shows the case when p=19, one time we get our function evaluates at 6520646525, and another time we see 5. Problems like this may take several tries until you are sure of the answer. Matt Last fiddled with by MattcAnderson on 2022-02-28 at 02:34 Reason: added second try. Code descrepancy
 2022-02-28, 04:32 #4 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 2×13×409 Posts Please check and make sure that you did not violate the magazine's copyright by copying this puzzle.
2022-02-28, 13:22   #5
Dr Sardonicus

Feb 2017
Nowhere

2·3·983 Posts

Quote:
 Originally Posted by MattcAnderson oops You are right, those 'p' values are composite numbers. At least I tried a hard problem.
"Hard problem?" Nonsense. You said yourself it's in the "quickies" section.
Quote:
 My claim is that there are five prime solutions for p. They are 2,3,5,11, and 19. I searched p to 10,000. I would like confirmation from someone else.
OK. I will confirm that, other than p = 2, 3, 5, 11, and 19, there are no other primes p < 10000 for which p^4-35*p^3+365*p^2-1225*p+1259 is prime.
Quote:
 Also, close inspection shows the case when p=19, one time we get our function evaluates at 6520646525, and another time we see 5.
FWIW, 6520646525 is the evaluation of p^4-35*p^3+365*p^2-1225*p+1259 at p = 293.

 2022-03-01, 03:38 #6 axn     Jun 2003 5,387 Posts A quick check shows that its a mystery Last fiddled with by axn on 2022-03-01 at 14:06 Reason: camouflage

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