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Message boards : General discussion : ID that identity

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ID: 55391
Credit: 888,213,209
RAC: 138,415
Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (34,593,437)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (45,081,394)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (463,452,443)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (4,436,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 153992 - Posted: 7 Feb 2022 | 3:03:44 UTC
Last modified: 7 Feb 2022 | 3:43:44 UTC

n - 1 ----- \ i n X k = (k - 1) / (k - 1) / ----- i = 0

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Joined: 16 Feb 10
Posts: 1022
ID: 55391
Credit: 888,213,209
RAC: 138,415
Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (34,593,437)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (45,081,394)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (463,452,443)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (4,436,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 154009 - Posted: 7 Feb 2022 | 14:53:49 UTC

I think it doesn't have a name.
It's just the result of the polynomial division of (x^n - 1) / (x - 1)
which happens to have terms for the complete set of non-negative powers of x less than n,
which is what the summation indicates.

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Joined: 16 Feb 10
Posts: 1022
ID: 55391
Credit: 888,213,209
RAC: 138,415
Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (34,593,437)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (45,081,394)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (463,452,443)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (4,436,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 154014 - Posted: 7 Feb 2022 | 20:38:59 UTC

It has a form, if not a name.

It's the partial sum (sum of the first n terms) of the geometric series with constant ratio k.

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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (16,540,110)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,228,147)WW Turquoise: Earned 5,000,000 credits (9,640,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 154017 - Posted: 7 Feb 2022 | 22:05:46 UTC

Correct, see e.g. Wikipedia: Geometric series ยง Sum. /JeppeSN

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Joined: 16 Feb 10
Posts: 1022
ID: 55391
Credit: 888,213,209
RAC: 138,415
Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (34,593,437)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (45,081,394)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (463,452,443)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (4,436,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 154020 - Posted: 8 Feb 2022 | 0:12:59 UTC
Last modified: 8 Feb 2022 | 0:16:10 UTC

Thanks for that confirmation. I appreciate it.

I've known the answer for k = 2 intimately well since I worked it out by hand over 40 years ago.
The sum 2i from i = 0 to i = (n-1), is 2n - 1.

Today I wondered if there is such a formula for k = 3.
After playing a bit (using awesomely more advanced tools) I found
2(sum 3i from i = 0 to i = n-1) = 3n - 1.
Which, oddly enough, I've seen before in the form of a recurrence relation.

With a few more minutes of play I determined that
(k-1)(sum of ki from i = 0 to i = n-1) = kn - 1
and convinced myself that this formula is correct, without proof
(or technically, I used a probabilistic proof, by plugging in multiple
large values for k and n and always receiving a result of equality).

k = 1 is a special case which works in the multiplication form (0)(k) = 0,
but leads to division by zero in the polynomial division form (k^n-1)/(k-1)
So the closed form solution to that summation requires k > 1.

To answer the question I posed here, I searched for "sums of powers" but
search engines invariably return hits for "power sums" which are mathematically
more interesting (Zeta functions) since these are objects of modern research,
whereas geometric series have been beat to death since the time of Classical Greek philosophers.

Forgetting that once-upon-a-time I could recognize these sums as geometric series,
I was having an atypically hard time finding a relevant answer until I saw a post in
one of the question-and-answer forums. By then I had already created this thread.

So the main finding here is obvious: search engines are not oracles.
Web searches are biased against holders of zero knowledge.
To get a proper answer from a search engine you must at know the correct search term,
or at least know that what you are finding is not what you are looking for. In that case you must
consult an expert.

In this context an expert is anyone who has more than zero knowledge.
That's what the post in the question-and-answer forum turned out to be. Then knowing the
search term "geometric series" would be more productive, I could get my customarily precise
answers from a search engine.

It's easy to see why the average person is not excited by seach engines.
They are often clueless about what they are searching for, so they get
whatever drivel is being pushed by profit-motivated individuals,
including, unremarkably, the search engines themselves.
I see those too but just long enough to skip down past advertisements.

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Message boards : General discussion : ID that identity

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