a puzzle from 'mathematics magazine'

MattcAnderson · 2022-02-27, 16:57

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

Dr Sardonicus · 2022-02-27, 19:51

(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}
<snip>

Your reading skills leave much to be desired.

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

MattcAnderson · 2022-02-28, 02:21

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

Uncwilly · 2022-02-28, 04:32

Please check and make sure that you did not violate the magazine's copyright by copying this puzzle.

Dr Sardonicus · 2022-02-28, 13:22

Quote:

Originally Posted by MattcAnderson

oops
You are right, those 'p' values are composite numbers.
At least I tried a hard problem.
<sdnip>

"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.<snip>

FWIW, 6520646525 is the evaluation of p^4-35*p^3+365*p^2-1225*p+1259 at p = 293.

axn · 2022-03-01, 03:38

A quick check shows that its a mystery

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^4-35p^3+365p^2-1225p+1259); 4 3 2 p - 35 p + 365 p - 1225 p + 1259 > ifactor(1225); 2 2 (5) (7) > ifactor(1259); (1259) > c := p^4-35p^3+365p^2-1225p+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^4-35b^3+365b^2-1225b+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*

2022-03-01, 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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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.