Problems & Puzzles: Puzzles

 Puzzle 978. Improve this curio Carlos Rivera asks this: Blanchette found that the sum of the first 3618 primes, prime squares, prime cubes and primes to the fourth power are each prime values. Q. Improve the record by Blanchette: the sum of the first k primes, squared primes, cubed primes, fourth power primes and fifth power primes, are each prime values, for the same k.

The following contributions came all during the week 9-15, November 2019: Oscar Volpatti, Shyam Sunder Gupta, Seiji Tomita, Adam Stinchcombe, Paul Cleary, Metin Sariyar, Gennady Gusev, Simon Cavegn

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Oscar wrote:

Considering the first k = 42060710 primes, the sum of their n-th powers is a prime for 1<=n<=5:

s1 = 16750223092850711,
s2 = 9064056565293560536244137,
s3 = 5542379777022131768657116060449197,
s4 = 3621159527531854892055249532396340526866869,
s5 = 2466592608886758186821780328594711124301045269323221,
s6 = 1728984737870447317210937316968092906790022799799937136090597,
s6 = 13*17*103387*10743511*1598487277*451675546567577*9755531981311168123369.

The sum of the n-th powers of the first 5167280132 primes is a prime for 1<=n<=6.

s1 = 320377357694121826139,
s2 = 26877016989249602488079926691527,
s3 = 2545463008145740961828787151915009541643281,
s4 = 257498790990162090525554973097137749100991149524996371,
s5 = 27152276510104442356177601374988624303902242725841780997235944409,
s6 = 2946039574326471089259841684023685216615836705288986173286075639737721000707,
s7 = 326384500158608867440741223062333317848390375560562035043815125629440527829306788331241,
s7 = 13 * 251 * 499 * 5557 * 2885622167 * 12500634131097380237035284400836298628597011302436278856092398539247.

***

Shyam wrote:

The smallest solution I found is k = 42060710 that is
the sum of the first 42060710 primes, squared primes, cubed primes, fourth power primes and fifth power
primes, are each prime values, for the same k.
42060710th prime is 818752171

The other solutions obtained by me excluding k = 42060710 are
k = 68652990, 74290388, 76351112, 104550750

***

Paul wrote:

An update

Here is the next solution to the original puzzle.

743322 , {4040783433367, 30031255245272979989, 252532250549642397306369349, 2270133838344992974548404138216921}.

As a matter of interest here are all the Solutions to Blanchette’s original problem for all primes below 10^9.

3618                 743322              1557552
1808652            2128724            2242290
2486874            3534560            4220258
4617614            4763114            5701340
5894718            5921088            6378312
8376932            8883662            9021482
10113080         10766654         11172798
12168908         14188014         14806460
16369448         19775468         19954778
21259494         21855360         22533638
22630404         22864470         23598584
23843570         25947242         26089992
26714172         27053724         27498822
27889508         29843210         30520364
30707982         30954120         30972914
31141124         32606880         32755340
33886898         33982634         34885104
37109688         38406168         40529322
40556930         41940954         42060710
42568710         42626940         43033802
43061438         43218944         43682784
44954730         45932982         45950508
47937180         48052788         49179548
49410998         50357634         50808804
51854222         52214160         52513394
52668384         53179254         53276172
53704848         56518220         57139370
58269374         59309010         59552030
59708070         59735450         63254360
64406762         65283408         65978592
67892220         68652990         69296670
69439110         69963032         69994452
70260614         71223890         72551024
73738628         73796984         74290388
74369820         75204762         75528090
76351112         76486304         77213148
78273198         79595370         79921254
80476184         81466422         82132022
82435950         82454204         82590102
83490278         83774390         86891570
88799132         89012400         89153442
89406530         89878748         91550162
93032558         93380162         99904374

***

Seiji wrote:

First 42060710 primes.
n  sum of prime^n
[1, 16750223092850711]
[2, 9064056565293560536244137]
[3, 5542379777022131768657116060449197]
[4, 3621159527531854892055249532396340526866869]
[5, 2466592608886758186821780328594711124301045269323221]

First 5167280132 primes.
n  sum of prime^n
[1, 320377357694121826139]
[2, 26877016989249602488079926691527]
[3, 2545463008145740961828787151915009541643281]
[4, 257498790990162090525554973097137749100991149524996371]
[5, 27152276510104442356177601374988624303902242725841780997235944409]
[6, 2946039574326471089259841684023685216615836705288986173286075639737721000707]

***

I woke up thinking I would report "no findings" and found on my computer, p=818752171 has power of all primes sums equal to a prime for powers 1,2,3,4,5.  I searched for powers up to the tenth, and found other curios, like all odd powers 1,3,5,7,9 for p=2749723, and five consecutive powers for powers 4,5,6,7,8 with p=80990639 (but not starting with 1).

***

Metin wrote:

The sum of first k primes, squared primes, cubed primes, fourth power primes and fifth power primes are each prime values for the k I could find are;

42060710
68652990
74290388
76351112

***

#prime: 42060710,
prime: 818752171,
sum of primes: 16750223092850711,
sum of squared primes: 29064056565293560536244137,
sum of cubed primes: 5542379777022131768657116060449197,
sum of fourth power primes: 3621159527531854892055249532396340526866869,
sum of fifth power primes: 2466592608886758186821780328594711124301045269323221

***

Simon wrote:

n=1 k=1 sums:
2

n=2 k=2 sums:
5
13

n=3 and n=4 k=3618 sums:
57584381,
1276400440541,
32129765754996971,
865842131651904557537

n=5 k=42060710 sums:
16750223092850711,
9064056565293560536244137,
554237977702213176865711606044
9197,
362115952753185489205524953239
6340526866869,
246659260888675818682178032859
4711124301045269323221

n=6 k=5167280132 sums:
320377357694121826139,
268770169892496024880799266915
27,
254546300814574096182878715191
5009541643281,
257498790990162090525554973097
137749100991149524996371,
271522765101044423561776013749
88624303902242725841780997235944409,
294603957432647108925984168402
3685216615836705288986173286075639737721000707

***

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