The following code computes lower bounds for the smallest possible such $c$.
- After 15 minutes of computation, my laptop got that $c>41$ (see the details below),
- After 18 hours, it got that $c>59$ (improveable by more computation time).
Here is how the code works: assume that such $c$ exists. We know that $\lfloor c\rfloor$ is prime, so $c \ge 2$. Then $c^2 \ge 4$, but $\lfloor c^2 \rfloor$ is prime, so $c^2 \ge 5$ and then $c \ge 5^{1/2}$. Now, $\lfloor 5^{1/2} \rfloor = 2$, $\lfloor 5^{2/2} \rfloor = 5$, $\lfloor 5^{3/2} \rfloor = 11$, $\lfloor 5^{4/2} \rfloor = 25$. But $c \ge 5^{1/2}$, so $\lfloor c^4 \rfloor \ge \lfloor 5^{4/2} \rfloor = 25$ and $\lfloor c^4 \rfloor$ is prime, so $\lfloor c^4 \rfloor \ge 29$ (the next prime after $25$). Thus $c \ge 29^{1/4}$. Now $\lfloor 29^{1/4} \rfloor = 2$, $\lfloor 29^{2/4} \rfloor = 5$, $\lfloor 29^{3/4} \rfloor = 12$. But $c \ge 29^{1/4}$, so $\lfloor c^3 \rfloor \ge \lfloor 29^{3/4} \rfloor = 12$ and $\lfloor c^3 \rfloor$ is prime, so $\lfloor c^3 \rfloor \ge 13$ (the next prime after $12$). Thus $c \ge 13^{1/3}$...etc.
The computation provides also length records (see below for more details), for example:
- The smallest $x$ such that the sequence $(\lfloor x^n \rfloor)_n$ starts by $27$ prime numbers is $x = 38628691699209543232005302230980383283077723^{1/27} \simeq 41.146$,
- The smallest $x$ such that the sequence $(\lfloor x^n \rfloor)_n$ starts by $35$ prime numbers is $x = 4148748103580890708839735608204733043355735349912591106907117^{1/35} \simeq 53.94$.
___________
**Code**
sage: def pp(x,m,N):
....: if m==0:
....: return [x,N]
....: n=1
....: while is_prime(floor(x^n)):
....: n+=1
....: y=(next_prime(floor(x^n)))^(1/n)
....: if n>N:
....: print([y.n(),x,n])
....: if m>0:
....: return pp(y,m-1,max(N,n))
....:
___________
**Computations**
Details of the computation proving that $c>41$ (time = 15 minutes).
To get $c>59$ replace `range(50)` by `range(1022)` (see below, time = 18 hours).
*Prints explanation*: the computation prints two types of list:
- If it is of length $2$ then it corresponds to `[x.n(),[i]]` which means that after $2000i$ steps we reached the lower bound `x` (`x.n()` is just a numerical approximation).
- If it is of length $3$ then it corresponds to `[y.n(),x,n]` which means that for all $a < y$, the sequence $(\lfloor a^r \rfloor)_r$ starts by less than $n$ prime numbers, and $n-1$ is realized by $x<y$.
We took `m=2000` because if `m` is too big then SageMath crashed (on my laptop), then we iterated 50 times, so 100000 steps in all.
sage: [x,N]=[2,1]
....: for i in range(50):
....: print([x.n(),[i]])
....: [x,N]=pp(x,2000,N)
....:
[2.00000000000000, [0]]
[2.23606797749979, 2, 2]
[2.32059578710608, sqrt(5), 4]
[2.36051999685294, 31^(1/4), 6]
[2.36103991787210, 173^(1/6), 7]
[2.36144757900740, 409^(1/7), 8]
[2.36184719247834, 967^(1/8), 9]
[11.4754436180012, 3450844193^(1/9), 10]
[11.4772249460124, 39661481813^(1/10), 11]
[13.1754084688783, 2076849234433^(1/11), 12]
[13.9025228053976, 52134281654579^(1/12), 13]
[17.5369538773784, 14838980942616539^(1/13), 14]
[17.5369538773784, 260230524377962793^(1/14), 15]
[17.5369538773784, 4563650703502319197^(1/15), 16]
[17.5369538773784, 80032531899785490253^(1/16), 17]
[19.5947182358701, [1]]
[23.2708572819963, 172111744128569095516889^(1/17), 18]
[23.2708572819963, 4005187834171404283105501^(1/18), 19]
[23.4071193822604, [2]]
[29.2781355078466, [3]]
[29.2836275624131, 7342984643407766159814138311^(1/19), 20]
[29.2836275624130, 215029227494071397857756115239^(1/20), 22]
[29.2836275624130, 184394194768671251848277906031901^(1/22), 23]
[29.3140651944330, [4]]
[29.6863280163833, [5]]
[29.7199886617202, [6]]
[29.7262158681975, [7]]
[29.7986032159883, [8]]
[31.1080969548337, [9]]
[31.1656716760603, [10]]
[31.3602318150116, [11]]
[31.4936884203380, [12]]
[31.5855793146363, [13]]
[31.5877240204566, [14]]
[31.7708284209810, [15]]
[31.7776275935340, [16]]
[31.9228719777914, [17]]
[31.9561321469378, [18]]
[31.9561321470032, 40248230682190652963808004897577443^(1/23), 24]
[31.9561321470032, 1286177778363154944458206697482326941^(1/24), 25]
[37.0580160390721, [19]]
[37.0584389108062, [20]]
[37.0658517174746, [21]]
[37.1639162040819, [22]]
[37.1669125206849, [23]]
[37.1683288435420, [24]]
[37.1687268547021, [25]]
[37.4113116298099, [26]]
[37.4118531804157, [27]]
[37.4119519979104, 2114750864194724801026123348775078610409^(1/25), 26]
[37.4119519979104, 79116957818792486170093330814381247140659^(1/26), 27]
[37.4133160075168, [28]]
[37.5366773413800, [29]]
[37.5399875044127, [30]]
[37.5479790983487, [31]]
[37.5481223970385, [32]]
[37.7259910967192, [33]]
[37.7273014336926, [34]]
[37.7315275341604, [35]]
[37.7349245483495, [36]]
[37.7783534119867, [37]]
[37.7807097942423, [38]]
[37.7835286660001, [39]]
[37.7863629239001, [40]]
[37.8064776911507, [41]]
[37.8562779668531, [42]]
[37.8623613459024, [43]]
[37.9373990583452, [44]]
[37.9424408570224, [45]]
[37.9426346064883, [46]]
[37.9444485557837, [47]]
[41.1462253191129, 38628691699209543232005302230980383283077723^(1/27), 28]
[41.1521261606709, [48]]
[41.1533074278378, [49]]
The smallest $x$ such that the sequence $(\lfloor x^n \rfloor)_n$ starts by $27$ prime numbers is $x = 38628691699209543232005302230980383283077723^{1/27} \simeq 41.146$.
sage: x=38628691699209543232005302230980383283077723^(1/27)
sage: [factor(floor(x^n)) for n in range(1,28)]
[41,
1693,
69661,
2866289,
117936979,
4852661521,
199668704371,
8215613499241,
338041484174539,
13909131075052931,
572308241207202497,
23548323844696748659,
968924638801330588309,
39867591505359597096497,
1640400903009876073202443,
67496305168920571652572271,
2777218180688008234140354019,
114272045022925494706426972643,
4701863312189102352562450647343,
193463927263003124964953243458331,
7960310342283989048957005995760097,
327536722953681474755429159335036199,
13476899802936024915424299817959748633,
554523555894713462471785930643956846483,
22816551175599557614676776265536047443603,
938814955676288945033539543576440477832567,
38628691699209543232005302230980383283077723]
sage: x.n()
41.1462253191129
To get $c>59$ (time = 18 hours), here is the computation (we skipped the intermediate prints).
sage: [x,N]=[2,1]
....: for i in range(1022):
....: print([x.n(),[i]])
....: [x,N]=pp(x,2000,N)
....:
[2.00000000000000, [0]]
[2.23606797749979, 2, 2]
[2.32059578710608, sqrt(5), 4]
............... etc ....................
[53.9436697195599, 4148748103580890708839735608204733043355735349912591106907117^(1/35), 36]
[53.9436702309118, [1015]]
[53.9436784982251, [1016]]
[53.9436794835264, [1017]
[53.9436818954090, [1018]]
[53.9436819071867, [1019]]
[53.9436825661574, [1020]]
[59.0854858349532, [1021]]
The smallest $x$ such that the sequence $(\lfloor x^n \rfloor)_n$ starts by $35$ prime numbers is $x = 4148748103580890708839735608204733043355735349912591106907117^{1/35} \simeq 53.94$.
sage: x=4148748103580890708839735608204733043355735349912591106907117^(1/35)
sage: [factor(floor(x^n)) for n in range(1,36)]
[53,
2909,
156971,
8467631,
456775117,
24640126081,
1329178823209,
71700783437431,
3867803380382509,
208643508091551469,
11254996489620980821,
607135813330920533459,
32751133789239563874359,
1766716344067858106189647,
95303162952544927897525549,
5141002345541482903522204573,
277324532555372631935243953111,
14959902989298364748262440497297,
806992065891368174795394951094273,
43532113468749303653986100712896887,
2348281951152618500276459340180295249,
126674945981380601052391522154658777199,
6833311447762689695342149169976354827237,
368613895828998383350681753288959741868319,
19884386250639755929577516344783599870922529,
1072636764480669473521399910231762010493578667,
57861963352202618742765105036618265539241629887,
3121286640396498386560879368924457870106070776561,
168373655629623512662702993484676929764429005794377,
9082692868759332058734233410406296531041203742381329,
489953784276555631429428597665812242911507414311160479,
26429905116863028778863328677541914579343251077156126963,
1425726072343366036233932371293234811318245076148559248257,
76908896357055934782538327507552055223759094236947310955187,
4148748103580890708839735608204733043355735349912591106907117]
sage:
sage: x.n()
53.9436697195599