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-   -   Sums of all Squares (https://www.mersenneforum.org/showthread.php?t=13181)

alexhiggins732 2010-04-12 17:17

[quote=flouran;211328]I guess Atkin is often mistaken for the diet :smile:[/quote]

[url]http://wwwhomes.uni-bielefeld.de/achim/prime_sieve.html[/url] has it labeled Sieve of Atkins and points the the PDF I referenced above.

Being a crank and all, its practically a given that I'll have many more misnomers than this :smile:.

Mini-Geek 2010-04-12 17:29

[quote=davieddy;211501]I thought a calory was 4.2 Joule, but nutritionists use
the term to refer to a kilocalory.

I don't think there are different types for different foods.
Even the "type of energy" it measures is the same for
everything.

David[/quote]
In America, the most commonly quoted rule is:
1 Calorie (with a capital C) = 1 kilocalorie = 1000 calories (with a lower-case c) = 4.2 kJ

davieddy 2010-04-12 17:59

Yes - I deliberated over the capitalization and singular
of calories. I don't think things were thoroughly standardized
pre SI.
I was weaned on the foot poundel, and considered the erg a
newfangled Napoleonic invention:smile:

David

PS is there any concensus about kB, KB, kb or Kb for killerbight?

xilman 2010-04-12 18:15

[quote=Mini-Geek;211512]In America, the most commonly quoted rule is:
1 Calorie (with a capital C) = 1 kilocalorie = 1000 calories (with a lower-case c) = 4.2 kJ[/quote]True in the UK too.

Though I'd phrase the relationship as 1000 calories is approximately 4.2kJ :smile:

Paul

cheesehead 2010-04-12 19:01

Atkins sieve diet v1.1.2 bug-fix release
 
[quote=cheesehead;211487]Well, the diet is a sieve, screening out [/quote]the calories borne by[quote]certain types of[/quote]food [quote], isn't it? :smile:[/quote]

davieddy 2010-04-12 19:11

Physical Units
 
I feel a new thread coming on, initiated by this latest
(hijacking) exchange.

David

Science and Technology would be the appropriate
location. Paul?

David

S485122 2010-04-12 20:25

[QUOTE=davieddy;211516]PS is there any concensus about kB, KB, kb or Kb for killerbight?[/QUOTE]Yes there is a standard. First of all the "k" should be lower case for all units, see the site of the BIPM that defines the SI system, more precisely the page [url=http://www.bipm.org/en/si/prefixes.html]SI prefixes[/url]... Except for 2[sup]10[/sup] (1024) bytes where it is KiB with a capital "K" an "i" to mark the power of two "kilo" of 1024 bytes. 10[sup]3[/sup] (1000) bytes is a kB. See the interesting discussion at [url=http://physics.nist.gov/cuu/Units/binary.html]Prefixes for binary multiples[/URL].

Then bi signifies bits and B signifies bytes.

So depending on the quantity you speak of : it should be 1 KiB for 1024 bytes and 1 kB for 1000 bytes.

1 kb represents 1000 bits.

KB and Kb are no units that I know of.

Jacob

davieddy 2010-04-12 22:01

THX Jacob.
I will either defer to my previous communique,
or get William Blipp on the job:smile:

David

only_human 2010-04-15 01:19

[quote=davar55;210978]Another way to extend this problem is to use
a base other than ten. I think binary.

2^2 + 3^2 + 5^2 + ... + p^2 = 2[sup]m[/sup]K

What is the smallest prime p such that
the sum of squares of all primes up to p
is a multiple of 2 (or 4 or 8 or 16 or ...).

This question can also be asked of first powers
or cubes of primes.

Since we basically compute in decimal or binary,
if there's an interesting number theoretic fact here
we may find it in one of these two related
sequences of sequences.[/quote]

Back to this part, I wanted to make a comment or two about sums of odd squares in binary that is too concrete to be nice mathematics but I have mused over recently. Speaking of odd primes (or any odd numbers actually), the squares are all 1 (mod 8), so a sum of these squares will include all these 1's added together as part of the result, therefore:

With regards to the sum of N odd squares (or in your case, squares of odd primes),
2[sup]m[/sup]K == N (mod 8)

I recently noticed that any odd square is one more than 8 times a triangular number:
n[sup]2[/sup] = 8t + 1

Trivial facts:
The nth triangular number is n(n+1)/2
Squares are the sum of N consecutive odd integers
Triangular Numbers are the sum of N consecutive integers
Two consecutive triangular numbers added together is equal to a square

Some aspect of these trivial facts might come in handy; calling a particular triangular number t[sub]i[/sub], and the next triangular number in sequence t[sub]i+1[/sub], we can look at twin primes and consider the sum of their squares:
p[sup]2[/sup] + (p+2)[sup]2[/sup] = 8t[sub]i[/sub] + 1 + 8t[sub]i+1[/sub] + 1
Since t[sub]i[/sub] + t[sub]i+1[/sub] is itself a square that means that the sum of the squares of a the pair twinned primes is two more than a eight times a square. I don't know a use for this except that it is elementary fact accessible to me and it applies to all consecutive odd numbers but of primes, only to twin primes because they are the only consecutive odd numbers that are also prime.

science_man_88 2010-04-20 14:05

[QUOTE=Mini-Geek;211512]In America, the most commonly quoted rule is:
1 Calorie (with a capital C) = 1 kilocalorie = 1000 calories (with a lower-case c) = 4.2 kJ[/QUOTE]
actually more exact it's 4.184 I think

as to the code if i know what to help with I do have a ASM forum I have gone to maybe they could turn it to straight asm ( apparently ASM is now changed with different OS but if we can get it to a form that will work on most OS'es would that be worthy ?)

science_man_88 2010-04-20 14:29

I know some assembly but not really enough to help plus to assemble anything I'll have to use my links to masm32 download, and possibly [url]http://win32assembly.online.fr/tut2.html[/url]


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