Problems & Puzzles: Puzzles

 Puzzle 934. The prime 7499 While solving our Puzzle 933, Jeff Heleen discovered this green emerald: the prime 7499, which has the following two properties needing not words: 7499^9 = 74994632695013731827926060475067499 7499^17 = 74999265676223797703801732050342536/                     0262553344102235443117350127499 Q. Find out more of these, prime or not, but if prime the better.

Contributions came from Fred Schneider, Paolo Lava, Simon Cavegn, Jeff Heleen, Emmanuel Vantieghem

***

Fred wrote on Dec 1, 2018:

I wrote a small Haskell program to find them.  You just need supply a proper cycle length.  (e.g. 8 for 43 and 7499, 2 for 251)

sol :: Int -> Int -> Int -> [Int]
sol n ub c = map fst \$ filter (\(x,l) -> l <= r && (floor (n' * 10.0 ** l) == n)) \$
map (\x -> (x, f \$ fromIntegral x)) [1..ub]
where r   = 10.0 ** (fromIntegral (floor (1.0 + log n' / log 10.0)) * (-1.0))
f x = f' x - (fromIntegral \$ floor \$ f' x)
where f' x = log n' * (c' * x) / log 10
n'  = fromIntegral n :: Double
c'  = fromIntegral c :: Double -- Length of cycle (e.g. 43's is 8)

It's fairly quick to find massive solutions.  I cherry-picked some primes found in puzzle 933 (including 7499).

Solutions for p = 2
for n where 2^(4*n + 1) is a green emerald and n < 100
[5,10,15,20,49,54,59,64,69,98]

Solutions for p = 43

for n where 43^(8*n+1) is a green emerald and n < 10000

[192,251,310,502,561,620,679,871,930,989,1181,1240,1299,1358,1550,1609,
1668,1727,1919,1978,2037,2229,2288,2347,2406,2598,2657,2716,2967,3026,
3085,3277,3336,3395,3454,3646,3705,3764,3956,4015,4074,4133,4325,4384,
4443,4694,4753,4812,5004,5063,5122,5181,5373,5432,5491,5683,5742,5801,
5860,6052,6111,6170,6229,6421,6480,6539,6731,6790,6849,6908,7100,7159,
7218,7469,7528,7587,7779,7838,7897,7956,8148,8207,8266,8458,8517,8576,
8635,8827,8886,8945,9196,9255,9314,9506,9565,9624,9683,9875,9934,9993]

Solutions for p = 7499

for n where 7499^(8*n+1) is a green emerald and n < 1000000
[1 **,2 **, 37274,37275,74547,74548,111821,111822,149094,149095,186367,186368,
186369,223641,223642,260914,260915,298188,298189,335461,335462,
372734,372735,410008,410009,447281,447282,484554,484555,484556,
521828,521829,559101,559102,596375,596376,633648,633649,670921,
670922,708195,708196,745468,745469,782741,782742,782743,820015,
820016,857288,857289,894562,894563,931835,931836,969108,969109]

** these correspond to 9 and 17 and are just included for completeness

Solutions for p = 251
for n where 251^(2*n+1) is a green emerald and n < 100000
[22372,22985,23598,24211,24824,25437,26050,26663,27276,27889,28502,
29115,29728,30341,30954,31567,32180,32793,33406,34019,34632,35245,
35858,36471,37084,37697,38310,38923,39536,40149,40762,41375,41988,
42601,43214,43827,44440,45053,45666,46279,46892,47505,48118,48731,
49344,49957,50570,51183,51796,52409,53022,53635,54248,54861,55474,
56087,56700,79685,80298,80911,81524,82137,82750,83363,83976,84589,
85202,85815,86428,87041,87654,88267,88880,89493,90106,90719,91332,
91945,92558,93171,93784,94397,95010,95623,96236,96849,97462,98075,
98688,99301,99914]

Solutions for p = 48751

for n where 48751^(10*n + 1) is a green emerald and n < 10m
[132593,179462,358924,405793,538386,585255,764717,811586,944179,
991048,1170510,1217379,1396841,1576303,1623172,1802634,1849503,
1982096,2028965,2208427,2255296,2387889,2434758,2614220,2661089,
2793682,2840551,3020013,3066882,3246344,3425806,3472675,3652137,
3699006,3831599,3878468,4057930,4104799,4237392,4284261,4463723,
4510592,4643185,4690054,4869516,4916385,5095847,5275309,5322178,
5501640,5548509,5681102,5727971,5907433,5954302,6086895,6133764,
6313226,6360095,6492688,6539557,6719019,6765888,6945350,7124812,
7171681,7351143,7398012,7530605,7577474,7756936,7803805,7936398,
7983267,8162729,8209598,8389060,8568522,8615391,8794853,8974315,
9021184,9200646,9247515,9380108,9426977,9606439,9653308,9785901,
9832770]

Solutions for p = 218749

for n where 218749^(16*n + 1) is a green emerald and n < 50m
[1850162,1885397,3735559,3770794,5656191,5691426,7541588,7576823,
9426985,9462220,11347617,11382852,13233014,13268249,15153646,
15188881,17039043,17074278,18924440,18959675,20845072,20880307,
22730469,22765704,22800939,24615866,24651101,24686336,26536498,
26571733,28421895,28457130,28492365,30307292,30342527,30377762,
32227924,32263159,34113321,34148556,34183791,35998718,36033953,
36069188,37919350,37954585,39804747,39839982,39875217,41725379,
41760614,41795849,43610776,43646011,45496173,45531408,45601878,
47381570,47416805,47452040,47487275,49302202,49337437,49372672]

Note: This is kind of interesting: Many of these are pairs which differ by 35235.

I checked 218759^(16*35235) and I see why:
9.9999878775574812672581042437521538205736275845722... × 10^3010447

Looking at it further.  The differences under 100m were of the form

1850162r + 35235s where r = 0 or 1, and s is between -4 and 1

So, it illustrates how multiplying by powers of p that are close to a power of 10 allow us to generate new green emerald solutions.

Solutions for p =
for n where  5422943^(4*n+ 1) is a green emerald and n < 100m
[2225965,16552119,18778084,33104238,49656357,51882322,54108287,66208476,
68434441,72886371,76082700,80534630,81790231,82760595,84986560,
87212525,89438490,91664455,92634819,97086749,98342350,99312714]

By the way, I used www.wolframalpha.com to check my initial solutions and later spot check.

As a postscript, I conjecture that all of the primes that are solutions in puzzle 933 have an infinite number of green emerald solutions

***

On Dec 3, 2018, Paolo wrote:

It appears that any prime p has at least one exponent k>1 such that p^k starts and ends in p.

For the first primes we have:
n      p       k

1   2   21

2   3   41

3   5   24

4   7   33

5   11   171

6   13   361

7   17   461

8   19   471

9   23   1281

10   29   1091

11   31   231

12   37   221

13   41   236

14   43   61

15   47   861

16   53   2761

17   59   241

18   61   546

19   67   3261

20   71   1991

21   73   6081

22   79   421

23   83   9541

24   89   5731

25   97   4461

26   101   1621

27   103   21501

28   107   10381

29   109   5051

30   113   1301

31   127   16301

32   131   30051

33   137   18601

34   139   13601

35   149   3171

36   151   8991

37   157   7561

38   163   3201

39   167   33501

40   173   8701

It appears again that primes have a lot of exponents (where some patters can be detected):

2: 21, 41, 61, 81, 101, 121, 141, 161, 197, 217, 237, 257, 277, 297, 317, 337, 357, 393, 413, 433, 453, 473, 493, 513, 533, 553, 589, 609, 629, 649, 669, 689, 709, 729, 749, 785, etc.

3: 41, 85, 129, 173, 217, 261, 305, 349, 389, 393, 433, 437, 477, 481, 521, 525, 565, 569, 609, 653, 697, 741, 785, 829, 873, 917, 961, 1001, 1005, 1045, etc.

5: 24, 34, 44, 54, 64, 74, 84, 94, 127, 137, 147, 157, 167, 177, 187, 220, 230, 240, 250, 260, 270, 280, 290, 313, 323, 333, 343, 353, 363, 373, 383, 416, 426,  etc.

...

43: 61, 533, 1005, 1537, 2009, 2481, 2541, 3013, 3485, 3957, 4017, 4489, 4961, etc.

47: 861, 1721, 5341, 6201, 9821, 10681, 15161, 16021, etc.

and so on.

It could be that  for 7499 the values 9 and 17 are not the only ones.

(So, Perhaps there is a proof of this or a counterxample?, CR)

In fact this is the question. I am still in search for the exponent for 211...

Found exponent for 211. It is 65151. No counterexamples at the moment. What is clear is that the search became harder and harder because some exponents are very high. 7499 is particular because of the low exponent (9).

***

Simon wrote on Dec 4, 2018:

Did not find a solution where p^e starts and ends with p for 6 digits up to exponent 1500.

p^e starts and ends with p (5 digits):
89499^2201  repeats when e = 1 mod 2200, last repetition: e = 4401 (2 Repetitions)
47687^2501  repeats when e = 1 mod 2500, last repetition: e = 5001 (2 Repetitions)
13568^2529  not repeating
11424^2751  does not repeat: 11424^5501 starts 11425, ends 11424
24999^2793  repeats when e = 1 mod 2792, last repetition: e = 39089 (14 Repetitions)
19193^3501  not repeating
89499^4401
47687^5001
16807^5101  not repeating

p^e starts and ends with p (4 digits):
7499^9
7499^17 p and e both prime
1157^301  repeats when e = 1 mod 300, last repetition: e = 14401 (48 Repetitions)
1157^601
4193^809  not repeating
1157^901
1157^1201
2401^1276  repeats when e = 1 mod 1275, last repetition: e = 56101 (44 Repetitions)
1157^1501
3376^1691  not repeating
1157^1801
1157^2101
4849^2301  repeats when e = 1 mod 2300, last repetition: e = 6901 (3 Repetitions)
1157^2401
2401^2551
1157^2701
1157^3001
2943^3109  not repeating
1157^3301
1157^3601
2401^3826
1157^3901
1711^4001  repeats when e = 1 mod 4000, last repetition: e = 24001 (6 Repetitions)
1557^4021  not repeating
7501^4197  repeats when e = 1 mod 4196, last repetition: e = 12589 (3 Repetitions)
1157^4201
2449^4451  not repeating
1157^4501
4933^4501  repeats when e = 1 mod 4500, last repetition: e = 18001 (4 Repetitions)
4849^4601  not repeating
1151^4651  p and e both prime, not repeating
1157^4801
1157^5101
2401^5101
1249^5179  p and e both prime, not repeating when e = 1 mod 5178, but when e = 1 mod 8378
1507^5301  not repeating
1157^5401
8693^5541  not repeating
1157^5701
2057^5789  repeats when e = 1 mod 5788, last repetition: e = 11577 (2 Repetitions)
1157^6001
2743^6201  repeats when e = 1 mod 6200, last repetition: e = 136401 (22 Repetitions)
1157^6301
1875^6305  repeats when e = 1 mod 6304, last repetition: e = 75649 (12 Repetitions)
2401^6376
1157^6601
1157^6901
4849^6901
1157^7201
2809^7251  not repeating
1157^7501
1193^7621  p and e both prime, repeats when e = 1 mod 7620, last repetition: e = 15241 (2 Repetitions)
2401^7651
1693^7685  repeats when e = 1 mod 7684, last repetition: e = 23053 (3 Repetitions)
2064^7751  not repeating
1157^7801
1711^8001
1157^8101
5057^8221  not repeating
1249^8379  repeats when e = 1 mod 8378, last repetition: e = 25135 (3 Repetitions)
1157^8401
7501^8393
2017^8501  p and e both prime, not repeating
9375^8635  repeats when e = 1 mod 8634, last repetition: e = 915205 (106 Repetitions)
1157^8701
2401^8926
1157^9001
4933^9001  p and e both prime
1157^9301
1157^9601
1157^9901
2293^9901  p and e both prime, not repeating
3111^10001  not repeating
1857^10101  not repeating
8801^10151  repeats when e = 1 mod 10150, last repetition: e = 30451 (3 Repetitions)
1157^10201
2401^10201
1157^10501
1157^10501
1157^10501
1157^10801
3125^11069  repeats when e = 1 mod 11068, last repetition: e = 30451 (3 Repetitions)
1251^11085  repeats when e = 1 mod 11084, last repetition: e = 177345 (16 Repetitions)
1157^11101
1157^11401
2401^11476
2057^11577
1157^11701
1157^12001
1711^12001
6809^12001  repeats when e = 1 mod 12000, last repetition: e = 72001 (6 Repetitions)
1157^12301
4375^12393  not repeating
2743^12401
1149^12501  not repeating
1157^12601
7501^12589
1875^12609
7807^12641  repeats when e = 1 mod 12640, last repetition: e = 25281 (2 Repetitions)
3943^12701  not repeating
2401^12751
2001^12831  not repeating
1157^12901
1232^13101  not repeating
1157^13201
1157^13501
4933^13501
1157^13801
2401^14026
1157^14101
1157^14401
4224^14851  not repeating
5001^15057  repeats when e = 1 mod 15056, last repetition: e = 617297 (41 Repetitions)
1376^15201  not repeating
1193^15241  p and e both prime, not repeating
2401^15301
1693^15369
1711^16001
3751^16239  repeats when e = 1 mod 16238, last repetition: e = 64953 (4 Repetitions)
3307^16261  repeats when e = 1 mod 16260, last repetition: e = 227641 (14 Repetitions)
2401^16576
1249^16757
5807^16773  repeats when e = 1 mod 16772, last repetition: e = 33545 (2 Repetitions)
9375^17269
2401^17851
4933^18001
9149^18201  repeats when e = 1 mod 18200, last repetition: e = 36401 (2 Repetitions)
1001^18431  repeats when e = 1 mod 18430, last repetition: e = 165871 (9 Repetitions)
2743^18601
1875^18913
1639^19001  repeats when e = 1 mod 19000, last repetition: e = 38001 (2 Repetitions)
2401^19126
5479^19251  repeats when e = 1 mod 19250, last repetition: e = 38501 (2 Repetitions)
4629^19501  repeats when e = 1 mod 19500, last repetition: e = 58501 (3 Repetitions)
9376^19549  repeats when e = 1 mod 19548, last repetition: e = 39097 (2 Repetitions)
1249^19957
1711^20001
8801^20301

A 4 digit number with multiple "unique" solutions:
1249^5179  no repetitions, both prime
1249^8379  3 repetitions
1249^16757  (8378*2+1)
1249^19957  17 repetitions up to 339253
1249^25135  (8378*3+1)
1249^28335  2 repetitions up to 56669
1249^36713 both prime, no repetitions
1249^39913 (19956*2+1)
1249^45091 no repetitions
1249^48291  2 repetitions up to 96581
1249^56669 (28334*2+1)
1249^59869 (19956*3+1)
1249^68247 2 repetitions up to 136493
1249^76625 no repetitions

A question is, can these 4 or 5 digit numbers be identified in another way?
I noticed most p^2 end either with 49 or 01.

...

(later)

...Here is a solution with 8 digits: 10095807^22440501

***

Jeff wrote on Dec 7, 2018:

Looking at primes p, 10 < p ≤ 25,000,000 for powers n from 2 to as high as I could go without an overflow error are the following:

 Prime, p powers, n 11 171, 291, 581, 871, 1161, 1451, 1741, 2031, 2321 13 361, 1221, 1581, 2081 17 461, 921, 1381 19 471, 801, 1271, 1601 23 1281 29 1091, 1171 31 231, 811, 1621 37 221, 441 41 236, 471, 1331    1566 43 61, 533, 1005, 1537 47 861 59 241 61 546, 1296 79 421 193 901 199 871 251 609 7499 9, 17

***

Emmanuel wrote on Dec 7, 2018:

I took  m  from  2  to  10^6  and  k  from  2  to  200  and looked if  m^k  began and ended in  m. Here is what I found :

m         k
2         {21, 41, 61, 81, 101, 121, 141, 161, 197}
3         {41, 85, 129, 173}
4         {11, 21, 31, 41, 99, 109, 119, 129, 139, 197}
5         {24, 34, 44, 54, 64, 74, 84, 94, 127, 137, 147, 157, 167, 177, 187}
6         {10, 19, 28, 37, 46, 55, 64, 73, 82, 91, 100, 109, 118, 127, 136, 145, 154, 163, 172}
7         {33, 117}
8         {73, 145, 197}
9         {153, 175, 197}
11       {171}
16       {6}
24       {51, 193}
25       {94, 187, 192}
32       {197}
36       {161}
43       {61}
49       {143}
51       {107}
56       {136}
75       {9, 17, 25, 33, 41, 49, 57, 65, 73, 81, 89}
76       {152}
624     {167}
7499   {9, 17}

For these  m  I counted the powers <= 10000  that ended and began with m :
m        #k
2         443
3         308
4         487
5         792
6         663
7         146
8         134
9         227
11        38
16        51
24        91
25        169
32        32
36        17
43        26
49        58
51        41
56        16
75        35
76        55
624      4
7499    2

This may suggest that there might be infinitely many  k  for some  m.

***

Jim Howell wrote on set 8, 2019

In the solutions to this puzzle, Fred Schneider found the exponents 8n+1 for n=37274 and 37275, corresponding to exponents 298193 and 298201.  I have found two smaller exponents which also solve the puzzle, namely 149101 and 149109.

7499^149101 = 749937584…..180757499 [577767 digits]

7499^149109 = 749983913…..655817499 [577798 digits]

I believe these are the smallest exponents which satisfy the puzzle, other than 9 and 17.

***

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