A little puzzle - Page 2

Dr Sardonicus · 2020-10-12, 14:24

Quote:

Originally Posted by R2357

Here goes, what do pi and the prime number sequence have in common (apart from pi's usage in the prime counting function and in the Gapcoin symbol)?

Depends what you mean by "in common." The number pi and the sequence of primes are both mathematical concepts, so they have that in common.

Although not something "in common," the following identity does give a connection between pi and the primes.

$\zeta(2k)\;=\;\frac{(-1)^{k+1}B_{2k}(2\pi)^{2k}}{2(2k)!}$

where B_2k is a Bernoulli number.

Since the zeta values tend to 1 as k increases, this identity clearly indicates how fast the Bernoulli numbers increase.

By the Euler product formula,

$\zeta(2k)\;=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}\;=\prod_{p prime}(1\;-\;p^{-2k})^{-1}$

R2357 · 2020-10-12, 14:39

I meant that the last digit to occur in both, after all the others already occured is 0

kruoli · 2020-10-12, 14:45

Quote:

Originally Posted by R2357

What do the numbers 3, 4, 5, 9, 10 and 15 have in common?

A few days ago, inspired by your username, I implemented a search for those numbers, not knowing that 217040 already existed. I then tried a reverse search with my results, but it failed. So my results have two extra entries up to base 10,000:

Code:

Base   3: 2: PRIME
Base   4: 11: PRIME
Base   5: 13: PRIME
Base   9: 1753: PRIME
Base  10: 2357: PRIME
Base  15: 1689253: PRP
Base  30: 1381634376893: PRP
Base 244: 28053100326775557991040317463038978786222621912035656410548777019058410902967286473295967973013822496644381234187139444081933: PRP
Base 676: 5315512989556603852873960767599554802770217878668020639087299338491156364777624078216225481953314889906416405260185946717596817192222631893348352843296815623256539192116715852938204396771410495745825850415512817745323160064190785403596966251903413339560196993352297473618262854362429132930982141489639972093336215001054282483833934391405974661: PRP
Base 908: 774253167992829066544897022900803876521834801951506297558501436443754805658452155038606354553711608434818871357021885486538136267223941008557077136840630016149444258651378366264186840692518095028210114290149037111207053828312230815317283218154106634363766516912291180016347613190975638154063456737387762950562305089646451132355346897253623075248726702532368645356588615208550106471096051303922403902187226965975405164795025681172388691682609981655338411: PRP

Are these the correct values or is OEIS wrong? The ones marked as "PRP" are proven primes in FactorDB.

Dr Sardonicus · 2020-10-12, 14:48

Quote:

Originally Posted by R2357

I meant that the last digit to occur in both, after all the others already occured is 0

Not in base three, it isn't...

R2357 · 2020-10-12, 14:58

Quote:

Originally Posted by kruoli

A few days ago, inspired by your username, I implemented a search for those numbers, not knowing that 217040 already existed. I then tried a reverse search with my results, but it failed. So my results have two extra entries up to base 10,000:

Code:

Base   3: 2: PRIME
Base   4: 11: PRIME
Base   5: 13: PRIME
Base   9: 1753: PRIME
Base  10: 2357: PRIME
Base  15: 1689253: PRP
Base  30: 1381634376893: PRP
Base 244: 28053100326775557991040317463038978786222621912035656410548777019058410902967286473295967973013822496644381234187139444081933: PRP
Base 676: 5315512989556603852873960767599554802770217878668020639087299338491156364777624078216225481953314889906416405260185946717596817192222631893348352843296815623256539192116715852938204396771410495745825850415512817745323160064190785403596966251903413339560196993352297473618262854362429132930982141489639972093336215001054282483833934391405974661: PRP
Base 908: 774253167992829066544897022900803876521834801951506297558501436443754805658452155038606354553711608434818871357021885486538136267223941008557077136840630016149444258651378366264186840692518095028210114290149037111207053828312230815317283218154106634363766516912291180016347613190975638154063456737387762950562305089646451132355346897253623075248726702532368645356588615208550106471096051303922403902187226965975405164795025681172388691682609981655338411: PRP

Are these the correct values or is OEIS wrong? The ones marked as "PRP" are proven primes in FactorDB.

I don't know about the last one but fore base 30, I checked and it seems that OEIS is right, it's not prime.

R2357 · 2020-10-12, 15:00

How did you get the result for b=30, I didn't get the same.

R2357 · 2020-10-12, 15:02

Quote:

Originally Posted by Dr Sardonicus

Not in base three, it isn't...

Sorry, I should have written base 10

R2357 · 2020-10-12, 15:13

Quote:

Originally Posted by sweety439

They are all divisors of 360 (the number of degrees in a turn)

Yes, but that wasn't what I had in mind, it's already been found.

R2357 · 2020-10-12, 15:29

Kruoli, for b=30, I found 41449031306819

Dr Sardonicus · 2020-10-12, 15:34

Quote:

Originally Posted by kruoli

<snip>

Code:

Base  30: 1381634376893: PRP
<snip>
Base 908: 77425316799...

<snip>

For base 30, you only concatenated the primes up to 23. You missed the prime 29.

Likewise for 908, you missed the prime 907.

kruoli · 2020-10-12, 16:05

Thanks, I found the error in my code...

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2020-10-12, 14:39	#13
R2357 "Ruben" Oct 2020 Nederland 2·19 Posts	I meant that the last digit to occur in both, after all the others already occured is 0

2020-10-12, 15:00	#17
R2357 "Ruben" Oct 2020 Nederland 2·19 Posts	How did you get the result for b=30, I didn't get the same.

2020-10-12, 15:29	#20
R2357 "Ruben" Oct 2020 Nederland 2×19 Posts	Kruoli, for b=30, I found 41449031306819

2020-10-12, 16:05	#22
kruoli "Oliver" Sep 2017 Porta Westfalica, DE 23·71 Posts	Thanks, I found the error in my code...