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Rich Holme
science forum beginner

Joined: 06 Jun 2005
Posts: 45

Posted: Thu Mar 24, 2005 7:53 pm    Post subject: Re: Does 8.999999999..... equal 9?

Bob Harris <plasticnitlion@wrappermindspring.com> writes:

 Quote: Rich Holmes wrote: ... can you prove the assertion that 9.999... - 0.999... = 9? Or is it 8.999...? Yes

Perhaps you misunderstood my point. The suggested proof makes use of
decimal arithmetic (to infinite precision) on infinitely repeating
decimal fractions. Conventional arithmetic procedures on decimal
numbers are algorithms that work on one decimal place at a time; they
will never terminate, meaning they will give no final result, when
applied to infinitely repeating fractions. In order for the proof to
work, one needs to supply proof, in some form other than an arithmetic
algorithm, that 9.999... - 0.999... = 9. Of course this is a true
statement (as is the statement 9.999... - 0.999... = 8.999...), but
one cannot simply assert it to be true.
Timothy Little
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 295

Posted: Thu Mar 24, 2005 7:53 pm    Post subject: Re: Does 8.999999999..... equal 9?

Bob Harris wrote:
 Quote: I was answering an either/or question with a "yes". Was I trying to make an intelligent contribution to the discussion? Or was I just trying to be a smart ass?

Yes. Obviously.

- Tim
Bob Pease
science forum beginner

Joined: 29 Apr 2005
Posts: 47

Posted: Thu Mar 24, 2005 7:53 pm    Post subject: Re: A day of a CAS super-hero

"Rainer Rosenthal" <r.rosenthal@web.de> wrote in message
news:3a4vmbF66df77U1@individual.net...
 Quote: "Alec Mihailovs" wrote That is a good demonstration of one specific weakness of automated bug testing. Most of the bugs that were found that way are such bugs that nobody meets in real calculations and nobody cares about (except automated bug testing system creator). ... and here we enter the recreational part! There is an old joke of mine, which I am quite proud of: Two programs meet. Says the one: "Do you still believe in programmers?"

Two programs meet.
Says the one:
"Do you still believe in programmers?"

says the other

" I used to, but I don't anymore, as I became an Agnostic and lost
faith in etaoin shrdlu!!"

RJ Pease
Rainer Rosenthal
science forum Guru Wannabe

Joined: 29 Apr 2005
Posts: 114

Posted: Thu Mar 24, 2005 8:37 pm    Post subject: Re: How long is 2^k mod n interesting?

"Rainer Rosenthal"

 Quote: Thanks! I looked up 3, 5, 11, 13, 19, 29, ... and THAT *looks* promising!

Well, indeed!
http://www.research.att.com/projects/OEIS?Anum=A001122
is full to the point. Periodicity pure :-)

I did reduce my table futher and out they sprang, these
"long primes" as they are called in the "Book of Numbers"
by Conway&Guy (p. 161 and pp. 169 ff).

I know of at least one researcher here (hi John!), who
will be happy about further connections ... like that:

=========== from the comment on A001122 ===============
Pieter Moree writes (Oct 20 2004):
Assuming the Generalized Riemann Hypothesis it can
be shown that the density of primes p such that a
prescribed integer g has order (p-1)/t, with t fixed
exists and, moreover, it can be computed. This
density will be a rational number times the so called
Artin constant. For 2 and 10 the density of primitive
roots is A, the Artin constant itself.
=======================================================

Hmmm... in order to classify "my" numbers n with respect
to the behaviour of the sequence of residues of 2^k,
k=0,1,2,..., I did introduce a simple notation. And I
keep the residues r small: - n/2 < r <= n/2.

Take case n=36 for example, where the sequence of residues

1, 2, 4, 8, 16, -4, ...

Everything is said here, repeated doubling won't give
exciting news any more: -4, -8, -16, 4, 8, 16, -4, ...
is all we can expect. This is 3 terms followed by their
negatives. And a starting sequence of 2 terms "1,2".

Short: [2,3,m] (m for "minus")

And now take n=28 as another example, where the sequence

1, 2, 4, 8, -12, 4, 8, -12, 4, ...

Here we have a starting sequence "1, 2" of two terms again
and we have three terms for the cycle again: "4,8,-12", but
this time there is no m = minus, but a p = plus, because
now the terms repeat without changed sign.

Short: [2,3,p] (p for "plus")

For n = power of two like for example n = 8, we have

1, 2, 4, 0, 0, 0, 0, 0, 0, ...

with a starting sequence "1,2,4" of length 3 and a single
element in the loop, which in my opinion deserves neither
an "m" nor a "p" but an "o" (matter of taste, I believe):

Short: [3,1,o] (o for "nullify")

The "long primes" in this notation are:

n: 3 5 11 13 19
--------------------------------------------------------
Short: [0,1,m] [0,2,m] [0,5,m] [0,6,m] [0,9,m]

the next ones are 29 = [0,14,m] and 37 = [0,18,m].
I like my table of residues Table(n,k) = 2^k mod n
and the short notations. I will have to look for quite
a while though, until I will be able to *see* these
meganumbers n with 2^n = 3 (mod n).

Best regards,
Rainer Rosenthal
r.rosenthal@web.de
John Gabbriel
science forum beginner

Joined: 29 Apr 2005
Posts: 23

Posted: Fri Mar 25, 2005 6:44 am    Post subject: Re: Given a plane convex closed curve

Bob Pease wrote:
 Quote: astanoff@yahoo.fr> wrote in message news:1111663342.606914.272940@f14g2000cwb.googlegroups.com... Thank you for clearing what was confused : "circumference/greatest diameter <= pi " that was exactly what i meant but could not state correctly maybe due to my poor english ! It seems to be a difficult problem to me.

This follows from Barbier's theorem. though:

All convex curves of constant width d have perimeter pi*d.

Say the diameter of the convex closed curve is d. With A,B on the curve
being d apart. Draw a tangent to the curve at A. Draw a line segement
AX such that AX is of length d. Now move the point of intersection of
the Tangent and line segment (assuming that the tangent and the line
segment are rigid rods) from A to B along one arc and plot the locus of
X. The arc along which we moved plus the locus of X is a closed curve
of width d, which 'circumscribes' our curve. Thus perimeter of our
curve <= perimeter of new curve = pi*d.

Don't know how difficult the proof of Barbier's theorem is.

Maybe an elementary/easier proof would involve proving it for simple
closed convex polygons. (The result would then follow by taking a
limiting set of polygons...)
Rainer Rosenthal
science forum Guru Wannabe

Joined: 29 Apr 2005
Posts: 114

Posted: Fri Mar 25, 2005 7:22 am    Post subject: Re: How long is 2^k mod n interesting?

"Rainer Rosenthal"

 Quote: The "long primes" in this notation are: n: 3 5 11 13 19 -------------------------------------------------------- Short: [0,1,m] [0,2,m] [0,5,m] [0,6,m] [0,9,m] the next ones are 29 = [0,14,m] and 37 = [0,18,m]. I like my table of residues Table(n,k) = 2^k mod n and the short notations. I will have to look for quite a while though, until I will be able to *see* these meganumbers n with 2^n = 3 (mod n).

This is my table, the "long primes" marked with an asterisk:

n= 2: [ 1, 1,o]
n= 3: [ 0, 1,m]*
n= 4: [ 2, 1,o]
n= 5: [ 0, 2,m]*
n= 6: [ 1, 1,m]
n= 7: [ 0, 3,p]
n= 8: [ 3, 1,o]
n= 9: [ 0, 3,m]
n= 10: [ 1, 2,m]
n= 11: [ 0, 5,m]*
n= 12: [ 2, 1,m]
n= 13: [ 0, 6,m]*
n= 14: [ 1, 3,p]
n= 15: [ 0, 4,p]
n= 16: [ 4, 1,o]
n= 17: [ 0, 4,m]
n= 18: [ 1, 3,m]
n= 19: [ 0, 9,m]*
n= 20: [ 2, 2,m]
n= 21: [ 0, 6,p]
n= 22: [ 1, 5,m]
n= 23: [ 0, 11,p]
n= 24: [ 3, 1,m]
n= 25: [ 0, 10,m]
n= 26: [ 1, 6,m]
n= 27: [ 0, 9,m]
n= 28: [ 2, 3,p]
n= 29: [ 0, 14,m]*
n= 30: [ 1, 4,p]
n= 31: [ 0, 5,p]
n= 32: [ 5, 1,o]
n= 33: [ 0, 5,m]
n= 34: [ 1, 4,m]
n= 35: [ 0, 12,p]
n= 36: [ 2, 3,m]
n= 37: [ 0, 18,m]*
n= 38: [ 1, 9,m]
n= 39: [ 0, 12,p]
n= 40: [ 3, 2,m]
n= 41: [ 0, 10,m]
n= 42: [ 1, 6,p]
n= 43: [ 0, 7,m]
n= 44: [ 2, 5,m]
n= 45: [ 0, 12,p]
n= 46: [ 1, 11,p]
n= 47: [ 0, 23,p]
n= 48: [ 4, 1,m]
n= 49: [ 0, 21,p]
n= 50: [ 1, 10,m]
n= 51: [ 0, 8,p]
n= 52: [ 2, 6,m]
n= 53: [ 0, 26,m]*
n= 54: [ 1, 9,m]
n= 55: [ 0, 20,p]
n= 56: [ 3, 3,p]
n= 57: [ 0, 9,m]
n= 58: [ 1, 14,m]
n= 59: [ 0, 29,m]*
n= 60: [ 2, 4,p]
n= 61: [ 0, 30,m]*
n= 62: [ 1, 5,p]
n= 63: [ 0, 6,p]
n= 64: [ 6, 1,o]
n= 65: [ 0, 6,m]
n= 66: [ 1, 5,m]
n= 67: [ 0, 33,m]*
n= 68: [ 2, 4,m]
n= 69: [ 0, 22,p]
n= 70: [ 1, 12,p]
n= 71: [ 0, 35,p]
n= 72: [ 3, 3,m]
n= 73: [ 0, 9,p]
n= 74: [ 1, 18,m]
n= 75: [ 0, 20,p]
n= 76: [ 2, 9,m]
n= 77: [ 0, 30,p]
n= 78: [ 1, 12,p]
n= 79: [ 0, 39,p]
n= 80: [ 4, 2,m]
n= 81: [ 0, 27,m]
n= 82: [ 1, 10,m]
n= 83: [ 0, 41,m]*
n= 84: [ 2, 6,p]
n= 85: [ 0, 8,p]
n= 86: [ 1, 7,m]
n= 87: [ 0, 28,p]
n= 88: [ 3, 5,m]
n= 89: [ 0, 11,p]
n= 90: [ 1, 12,p]
n= 91: [ 0, 12,p]
n= 92: [ 2, 11,p]
n= 93: [ 0, 10,p]
n= 94: [ 1, 23,p]
n= 95: [ 0, 36,p]
n= 96: [ 5, 1,m]
n= 97: [ 0, 24,m]
n= 98: [ 1, 21,p]
n= 99: [ 0, 15,m]
n=100: [ 2, 10,m]
n=101: [ 0, 50,m]*
n=102: [ 1, 8,p]
n=103: [ 0, 51,p]
n=104: [ 3, 6,m]
n=105: [ 0, 12,p]
n=106: [ 1, 26,m]
n=107: [ 0, 53,m]*
n=108: [ 2, 9,m]
n=109: [ 0, 18,m]
n=110: [ 1, 20,p]
n=111: [ 0, 36,p]
n=112: [ 4, 3,p]
n=113: [ 0, 14,m]
n=114: [ 1, 9,m]
n=115: [ 0, 44,p]
n=116: [ 2, 14,m]
n=117: [ 0, 12,p]
n=118: [ 1, 29,m]
n=119: [ 0, 24,p]
n=120: [ 3, 4,p]
n=121: [ 0, 55,m]
n=122: [ 1, 30,m]
n=123: [ 0, 20,p]
n=124: [ 2, 5,p]
n=125: [ 0, 50,m]
n=126: [ 1, 6,p]
n=127: [ 0, 7,p]
n=128: [ 7, 1,o]
n=129: [ 0, 7,m]
n=130: [ 1, 6,m]
n=131: [ 0, 65,m]*
n=132: [ 2, 5,m]
n=133: [ 0, 18,p]
n=134: [ 1, 33,m]
n=135: [ 0, 36,p]
n=136: [ 3, 4,m]
n=137: [ 0, 34,m]
n=138: [ 1, 22,p]
n=139: [ 0, 69,m]*
n=140: [ 2, 12,p]
n=141: [ 0, 46,p]
n=142: [ 1, 35,p]
n=143: [ 0, 60,p]
n=144: [ 4, 3,m]
n=145: [ 0, 14,m]
n=146: [ 1, 9,p]
n=147: [ 0, 42,p]
n=148: [ 2, 18,m]
n=149: [ 0, 74,m]*
n=150: [ 1, 20,p]
n=151: [ 0, 15,p]
n=152: [ 3, 9,m]
n=153: [ 0, 24,p]
n=154: [ 1, 30,p]
n=155: [ 0, 20,p]
n=156: [ 2, 12,p]
n=157: [ 0, 26,m]
n=158: [ 1, 39,p]
n=159: [ 0, 52,p]
n=160: [ 5, 2,m]
n=161: [ 0, 33,p]
n=162: [ 1, 27,m]
n=163: [ 0, 81,m]*
n=164: [ 2, 10,m]
n=165: [ 0, 20,p]
n=166: [ 1, 41,m]
n=167: [ 0, 83,p]
n=168: [ 3, 6,p]
n=169: [ 0, 78,m]
n=170: [ 1, 8,p]
n=171: [ 0, 9,m]
n=172: [ 2, 7,m]
n=173: [ 0, 86,m]*
n=174: [ 1, 28,p]
n=175: [ 0, 60,p]
n=176: [ 4, 5,m]
n=177: [ 0, 29,m]
n=178: [ 1, 11,p]
n=179: [ 0, 89,m]*
n=180: [ 2, 12,p]
n=181: [ 0, 90,m]*
n=182: [ 1, 12,p]
n=183: [ 0, 60,p]
n=184: [ 3, 11,p]
n=185: [ 0, 18,m]
n=186: [ 1, 10,p]
n=187: [ 0, 40,p]
n=188: [ 2, 23,p]
n=189: [ 0, 18,p]
n=190: [ 1, 36,p]
n=191: [ 0, 95,p]
n=192: [ 6, 1,m]
n=193: [ 0, 48,m]
n=194: [ 1, 24,m]
n=195: [ 0, 12,p]
n=196: [ 2, 21,p]
n=197: [ 0, 98,m]*
n=198: [ 1, 15,m]
n=199: [ 0, 99,p]
n=200: [ 3, 10,m]
n=201: [ 0, 33,m]
n=202: [ 1, 50,m]
n=203: [ 0, 84,p]
n=204: [ 2, 8,p]
n=205: [ 0, 10,m]
n=206: [ 1, 51,p]
n=207: [ 0, 66,p]
n=208: [ 4, 6,m]
n=209: [ 0, 45,m]
n=210: [ 1, 12,p]
n=211: [ 0,105,m]*
n=212: [ 2, 26,m]
n=213: [ 0, 70,p]
n=214: [ 1, 53,m]
n=215: [ 0, 28,p]
n=216: [ 3, 9,m]
n=217: [ 0, 15,p]
n=218: [ 1, 18,m]
n=219: [ 0, 18,p]
n=220: [ 2, 20,p]
n=221: [ 0, 24,p]
n=222: [ 1, 36,p]
n=223: [ 0, 37,p]
n=224: [ 5, 3,p]
n=225: [ 0, 60,p]
n=226: [ 1, 14,m]
n=227: [ 0,113,m]*
n=228: [ 2, 9,m]
n=229: [ 0, 38,m]
n=230: [ 1, 44,p]
n=231: [ 0, 30,p]
n=232: [ 3, 14,m]
n=233: [ 0, 29,p]
n=234: [ 1, 12,p]
n=235: [ 0, 92,p]
n=236: [ 2, 29,m]
n=237: [ 0, 78,p]
n=238: [ 1, 24,p]
n=239: [ 0,119,p]
n=240: [ 4, 4,p]
n=241: [ 0, 12,m]
n=242: [ 1, 55,m]
n=243: [ 0, 81,m]
n=244: [ 2, 30,m]
n=245: [ 0, 84,p]
n=246: [ 1, 20,p]
n=247: [ 0, 36,p]
n=248: [ 3, 5,p]
n=249: [ 0, 41,m]
n=250: [ 1, 50,m]
n=251: [ 0, 25,m]
n=252: [ 2, 6,p]
n=253: [ 0,110,p]
n=254: [ 1, 7,p]
n=255: [ 0, 8,p]
n=256: [ 8, 1,o]
n=257: [ 0, 8,m]
n=258: [ 1, 7,m]
n=259: [ 0, 36,p]
n=260: [ 2, 6,m]
n=261: [ 0, 84,p]
n=262: [ 1, 65,m]
n=263: [ 0,131,p]
n=264: [ 3, 5,m]
n=265: [ 0, 26,m]
n=266: [ 1, 18,p]
n=267: [ 0, 22,p]
n=268: [ 2, 33,m]
n=269: [ 0,134,m]*
n=270: [ 1, 36,p]
n=271: [ 0,135,p]
n=272: [ 4, 4,m]
n=273: [ 0, 12,p]
n=274: [ 1, 34,m]
n=275: [ 0, 20,p]
n=276: [ 2, 22,p]
n=277: [ 0, 46,m]
n=278: [ 1, 69,m]
n=279: [ 0, 30,p]
n=280: [ 3, 12,p]
n=281: [ 0, 35,m]
n=282: [ 1, 46,p]
n=283: [ 0, 47,m]
n=284: [ 2, 35,p]
n=285: [ 0, 36,p]
n=286: [ 1, 60,p]
n=287: [ 0, 60,p]
n=288: [ 5, 3,m]
n=289: [ 0, 68,m]
n=290: [ 1, 14,m]
n=291: [ 0, 48,p]
n=292: [ 2, 9,p]
n=293: [ 0,146,m]*
n=294: [ 1, 42,p]
n=295: [ 0,116,p]
n=296: [ 3, 18,m]
n=297: [ 0, 45,m]
n=298: [ 1, 74,m]
n=299: [ 0,132,p]
n=300: [ 2, 20,p]
n=301: [ 0, 42,p]
n=302: [ 1, 15,p]
n=303: [ 0,100,p]
n=304: [ 4, 9,m]
n=305: [ 0, 30,m]
n=306: [ 1, 24,p]
n=307: [ 0, 51,m]
n=308: [ 2, 30,p]
n=309: [ 0,102,p]
n=310: [ 1, 20,p]
n=311: [ 0,155,p]
n=312: [ 3, 12,p]
n=313: [ 0, 78,m]
n=314: [ 1, 26,m]
n=315: [ 0, 12,p]
n=316: [ 2, 39,p]
n=317: [ 0,158,m]*
n=318: [ 1, 52,p]
n=319: [ 0,140,p]
n=320: [ 6, 2,m]
n=321: [ 0, 53,m]
n=322: [ 1, 33,p]
n=323: [ 0, 72,p]
n=324: [ 2, 27,m]
n=325: [ 0, 30,m]
n=326: [ 1, 81,m]
n=327: [ 0, 36,p]
n=328: [ 3, 10,m]
n=329: [ 0, 69,p]
n=330: [ 1, 20,p]
n=331: [ 0, 15,m]
n=332: [ 2, 41,m]
n=333: [ 0, 36,p]
n=334: [ 1, 83,p]
n=335: [ 0,132,p]
n=336: [ 4, 6,p]
n=337: [ 0, 21,p]
n=338: [ 1, 78,m]
n=339: [ 0, 28,p]
n=340: [ 2, 8,p]
n=341: [ 0, 10,p]
n=342: [ 1, 9,m]
n=343: [ 0,147,p]
n=344: [ 3, 7,m]
n=345: [ 0, 44,p]
n=346: [ 1, 86,m]
n=347: [ 0,173,m]*
n=348: [ 2, 28,p]
n=349: [ 0,174,m]*
n=350: [ 1, 60,p]
n=351: [ 0, 36,p]
n=352: [ 5, 5,m]
n=353: [ 0, 44,m]
n=354: [ 1, 29,m]
n=355: [ 0,140,p]
n=356: [ 2, 11,p]
n=357: [ 0, 24,p]
n=358: [ 1, 89,m]
n=359: [ 0,179,p]
n=360: [ 3, 12,p]
n=361: [ 0,171,m]
n=362: [ 1, 90,m]
n=363: [ 0, 55,m]
n=364: [ 2, 12,p]
n=365: [ 0, 36,p]
n=366: [ 1, 60,p]
n=367: [ 0,183,p]
n=368: [ 4, 11,p]
n=369: [ 0, 60,p]
n=370: [ 1, 18,m]
n=371: [ 0,156,p]
n=372: [ 2, 10,p]
n=373: [ 0,186,m]*
n=374: [ 1, 40,p]
n=375: [ 0,100,p]
n=376: [ 3, 23,p]
n=377: [ 0, 42,m]
n=378: [ 1, 18,p]
n=379: [ 0,189,m]*
n=380: [ 2, 36,p]
n=381: [ 0, 14,p]
n=382: [ 1, 95,p]
n=383: [ 0,191,p]
n=384: [ 7, 1,m]
n=385: [ 0, 60,p]
n=386: [ 1, 48,m]
n=387: [ 0, 21,m]
n=388: [ 2, 24,m]
n=389: [ 0,194,m]*
n=390: [ 1, 12,p]
n=391: [ 0, 88,p]
n=392: [ 3, 21,p]
n=393: [ 0, 65,m]
n=394: [ 1, 98,m]
n=395: [ 0,156,p]
n=396: [ 2, 15,m]
n=397: [ 0, 22,m]
n=398: [ 1, 99,p]
n=399: [ 0, 18,p]

Kind regards,
Rainer Rosenthal
r.rosenthal@web.de
Guest

 Posted: Fri Mar 25, 2005 8:13 am    Post subject: Re: Given a plane convex closed curve Thank you for your explanation. By the way, can the curve you use be called an "offset" curve? v.a.
Rainer Rosenthal
science forum Guru Wannabe

Joined: 29 Apr 2005
Posts: 114

Posted: Fri Mar 25, 2005 8:25 am    Post subject: Re: How long is 2^k mod n interesting?

"Rainer Rosenthal" wrote

 Quote: Hmmm... in order to classify "my" numbers n with respect to the behaviour of the sequence of residues of 2^k, k=0,1,2,..., I did introduce a simple notation. And I keep the residues r small: - n/2 < r <= n/2.

This is just another version of my table of residues and
classification [x,y,z], where each sequence of residues
is listed as long as (see posting subject) it is interesting.
Starting sequence and cycle are separated by semi-colon,
the leeter at the end is p=positive, if the next element
in the cycle is the same as the first one. It is m=minus,
if the next element in the cycle is the negative of the first
one. The letter o is used in the powers of two, because the
cycle consists of 0 alone.

For "long primes" with every residuum except 0 (marked with an
asterisk) the cycle always starts with 1 and has an m=minus,
which is logical, because -1 must be there, if every residuum
is in the cycle.

n= 2: [ 1, 1,o] (1;0)o
n= 3: [ 0, 1,m]* (;1)m
n= 4: [ 2, 1,o] (1,2;0)o
n= 5: [ 0, 2,m]* (;1,2)m
n= 6: [ 1, 1,m] (1;2)m
n= 7: [ 0, 3,p] (;1,2,-3)p
n= 8: [ 3, 1,o] (1,2,4;0)o
n= 9: [ 0, 3,m] (;1,2,4)m
n= 10: [ 1, 2,m] (1;2,4)m
n= 11: [ 0, 5,m]* (;1,2,4,-3,5)m
n= 12: [ 2, 1,m] (1,2;4)m
n= 13: [ 0, 6,m]* (;1,2,4,-5,3,6)m
n= 14: [ 1, 3,p] (1;2,4,-6)p
n= 15: [ 0, 4,p] (;1,2,4,-7)p
n= 16: [ 4, 1,o] (1,2,4,8;0)o
n= 17: [ 0, 4,m] (;1,2,4,m
n= 18: [ 1, 3,m] (1;2,4,m
n= 19: [ 0, 9,m]* (;1,2,4,8,-3,-6,7,-5,9)m
n= 20: [ 2, 2,m] (1,2;4,m
n= 21: [ 0, 6,p] (;1,2,4,8,-5,-10)p
n= 22: [ 1, 5,m] (1;2,4,8,-6,10)m
n= 23: [ 0, 11,p] (;1,2,4,8,-7,9,-5,-10,3,6,-11)p
n= 24: [ 3, 1,m] (1,2,4;m
n= 25: [ 0, 10,m] (;1,2,4,8,-9,7,-11,3,6,12)m
n= 26: [ 1, 6,m] (1;2,4,8,-10,6,12)m
n= 27: [ 0, 9,m] (;1,2,4,8,-11,5,10,-7,13)m
n= 28: [ 2, 3,p] (1,2;4,8,-12)p
n= 29: [ 0, 14,m]* (;1,2,4,8,-13,3,6,12,-5,-10,9,-11,7,14)m
n= 30: [ 1, 4,p] (1;2,4,8,-14)p
n= 31: [ 0, 5,p] (;1,2,4,8,-15)p
n= 32: [ 5, 1,o] (1,2,4,8,16;0)o
n= 33: [ 0, 5,m] (;1,2,4,8,16)m
n= 34: [ 1, 4,m] (1;2,4,8,16)m
n= 35: [ 0, 12,p] (;1,2,4,8,16,-3,-6,-12,11,-13,9,-17)p
n= 36: [ 2, 3,m] (1,2;4,8,16)m
n= 37: [ 0, 18,m]* (;1,2,4,8,16,-5,-10,17,-3,-6,-12,13,-11,15,-7,-14,9,1m
n= 38: [ 1, 9,m] (1;2,4,8,16,-6,-12,14,-10,1m
n= 39: [ 0, 12,p] (;1,2,4,8,16,-7,-14,11,-17,5,10,-19)p
n= 40: [ 3, 2,m] (1,2,4;8,16)m
n= 41: [ 0, 10,m] (;1,2,4,8,16,-9,-18,5,10,20)m
n= 42: [ 1, 6,p] (1;2,4,8,16,-10,-20)p
n= 43: [ 0, 7,m] (;1,2,4,8,16,-11,21)m
n= 44: [ 2, 5,m] (1,2;4,8,16,-12,20)m
n= 45: [ 0, 12,p] (;1,2,4,8,16,-13,19,-7,-14,17,-11,-22)p
n= 46: [ 1, 11,p] (1;2,4,8,16,-14,18,-10,-20,6,12,-22)p
n= 47: [ 0, 23,p]
(;1,2,4,8,16,-15,17,-13,21,-5,-10,-20,7,14,-19,9,18,-11,-22,
3,6,12,-23)p
n= 48: [ 4, 1,m] (1,2,4,8;16)m
n= 49: [ 0, 21,p]
(;1,2,4,8,16,-17,15,-19,11,22,-5,-10,-20,9,18,-13,23,-3,-6,
-12,-24)p
n= 50: [ 1, 10,m] (1;2,4,8,16,-18,14,-22,6,12,24)m
n= 51: [ 0, 8,p] (;1,2,4,8,16,-19,13,-25)p
n= 52: [ 2, 6,m] (1,2;4,8,16,-20,12,24)m
n= 53: [ 0, 26,m]*
(;1,2,4,8,16,-21,11,22,-9,-18,17,-19,15,-23,7,14,-25,3,6,12,
24,-5,-10,-20,13,26)m
n= 54: [ 1, 9,m] (1;2,4,8,16,-22,10,20,-14,26)m
n= 55: [ 0, 20,p]
(;1,2,4,8,16,-23,9,18,-19,17,-21,13,26,-3,-6,-12,-24,7,14,
-27)p
n= 56: [ 3, 3,p] (1,2,4;8,16,-24)p
n= 57: [ 0, 9,m] (;1,2,4,8,16,-25,7,14,2m
n= 58: [ 1, 14,m] (1;2,4,8,16,-26,6,12,24,-10,-20,18,-22,14,2m
n= 59: [ 0, 29,m]*
(;1,2,4,8,16,-27,5,10,20,-19,21,-17,25,-9,-18,23,-13,-26,7,
14,28,-3,-6,-12,-24,11,22,-15,29)m
n= 60: [ 2, 4,p] (1,2;4,8,16,-2p
n= 61: [ 0, 30,m]*
(;1,2,4,8,16,-29,3,6,12,24,-13,-26,9,18,-25,11,22,-17,27,-7,
-14,-28,5,10,20,-21,19,-23,15,30)m
n= 62: [ 1, 5,p] (1;2,4,8,16,-30)p
n= 63: [ 0, 6,p] (;1,2,4,8,16,-31)p
n= 64: [ 6, 1,o] (1,2,4,8,16,32;0)o
n= 65: [ 0, 6,m] (;1,2,4,8,16,32)m
n= 66: [ 1, 5,m] (1;2,4,8,16,32)m
n= 67: [ 0, 33,m]*
(;1,2,4,8,16,32,-3,-6,-12,-24,19,-29,9,18,-31,5,10,20,-27,13,
26,-15,-30,7,14,28,-11,-22,23,-21,25,-17,33)m
n= 68: [ 2, 4,m] (1,2;4,8,16,32)m
n= 69: [ 0, 22,p]
(;1,2,4,8,16,32,-5,-10,-20,29,-11,-22,25,-19,31,-7,-14,-28,13,
26,-17,-34)p
n= 70: [ 1, 12,p] (1;2,4,8,16,32,-6,-12,-24,22,-26,18,-34)p
n= 71: [ 0, 35,p]
(;1,2,4,8,16,32,-7,-14,-28,15,30,-11,-22,27,-17,-34,3,6,12,24,
-23,25,-21,29,-13,-26,19,-33,5,10,20,-31,9,18,-35)p
n= 72: [ 3, 3,m] (1,2,4;8,16,32)m
n= 73: [ 0, 9,p] (;1,2,4,8,16,32,-9,-18,-36)p
n= 74: [ 1, 18,m]
(1;2,4,8,16,32,-10,-20,34,-6,-12,-24,26,-22,30,-14,-28,18,36)m
n= 75: [ 0, 20,p]
(;1,2,4,8,16,32,-11,-22,31,-13,-26,23,-29,17,34,-7,-14,-28,
19,-37)p
n= 76: [ 2, 9,m] (1,2;4,8,16,32,-12,-24,28,-20,36)m
n= 77: [ 0, 30,p]
(;1,2,4,8,16,32,-13,-26,25,-27,23,-31,15,30,-17,-34,9,18,36,
-5,-10,-20,37,-3,-6,-12,-24,29,-19,-3p
n= 78: [ 1, 12,p] (1;2,4,8,16,32,-14,-28,22,-34,10,20,-3p
n= 79: [ 0, 39,p]
(;1,2,4,8,16,32,-15,-30,19,38,-3,-6,-12,-24,31,-17,-34,11,

22,-35,9,18,36,-7,-14,-28,23,-33,13,26,-27,25,-29,21,-37,5,
10,20,-39)p
n= 80: [ 4, 2,m] (1,2,4,8;16,32)m
n= 81: [ 0, 27,m]
(;1,2,4,8,16,32,-17,-34,13,26,-29,23,-35,11,22,-37,7,14,28,
-25,31,-19,-38,5,10,20,40)m
n= 82: [ 1, 10,m] (1;2,4,8,16,32,-18,-36,10,20,40)m
n= 83: [ 0, 41,m]*
(;1,2,4,8,16,32,-19,-38,7,14,28,-27,29,-25,33,-17,-34,15,30,
-23,37,-9,-18,-36,11,22,-39,5,10,20,40,-3,-6,-12,-24,35,-
13,
-26,31,-21,41)m
n= 84: [ 2, 6,p] (1,2;4,8,16,32,-20,-40)p
n= 85: [ 0, 8,p] (;1,2,4,8,16,32,-21,-42)p
n= 86: [ 1, 7,m] (1;2,4,8,16,32,-22,42)m
n= 87: [ 0, 28,p]
(;1,2,4,8,16,32,-23,41,-5,-10,-20,-40,7,14,28,-31,25,-37,13,
26,-35,17,34,-19,-38,11,22,-43)p
n= 88: [ 3, 5,m] (1,2,4;8,16,32,-24,40)m
n= 89: [ 0, 11,p] (;1,2,4,8,16,32,-25,39,-11,-22,-44)p
n= 90: [ 1, 12,p] (1;2,4,8,16,32,-26,38,-14,-28,34,-22,-44)p
n= 91: [ 0, 12,p] (;1,2,4,8,16,32,-27,37,-17,-34,23,-45)p
n= 92: [ 2, 11,p] (1,2;4,8,16,32,-28,36,-20,-40,12,24,-44)p
n= 93: [ 0, 10,p] (;1,2,4,8,16,32,-29,35,-23,-46)p
n= 94: [ 1, 23,p]
(1;2,4,8,16,32,-30,34,-26,42,-10,-20,-40,14,28,-38,18,36,-22,
-44,6,12,24,-46)p
n= 95: [ 0, 36,p]
(;1,2,4,8,16,32,-31,33,-29,37,-21,-42,11,22,44,-7,-14,-28,39,
-17,-34,27,-41,13,26,-43,9,18,36,-23,-46,3,6,12,24,-47)p
n= 96: [ 5, 1,m] (1,2,4,8,16;32)m
n= 97: [ 0, 24,m]
(;1,2,4,8,16,32,-33,31,-35,27,-43,11,22,44,-9,-18,-36,25,-47,
3,6,12,24,4m
n= 98: [ 1, 21,p]
(1;2,4,8,16,32,-34,30,-38,22,44,-10,-20,-40,18,36,-26,46,-6,
-12,-24,-4p
n= 99: [ 0, 15,m] (;1,2,4,8,16,32,-35,29,-41,17,34,-31,37,-25,49)m
n=100: [ 2, 10,m] (1,2;4,8,16,32,-36,28,-44,12,24,4m
n=101: [ 0, 50,m]*
(;1,2,4,8,16,32,-37,27,-47,7,14,28,-45,11,22,44,-13,-26,49,
-3,-6,-12,-24,-48,5,10,20,40,-21,-42,17,34,-33,35,-31,39,
-23,
-46,9,18,36,-29,43,-15,-30,41,-19,-38,25,50)m
n=102: [ 1, 8,p] (1;2,4,8,16,32,-38,26,-50)p
n=103: [ 0, 51,p]
(;1,2,4,8,16,32,-39,25,50,-3,-6,-12,-24,-48,7,14,28,-47,9,

18,36,-31,41,-21,-42,19,38,-27,49,-5,-10,-20,-40,23,46,-11,
-22,-44,15,30,-43,17,34,-35,33,-37,29,-45,13,26,-51)p
n=104: [ 3, 6,m] (1,2,4;8,16,32,-40,24,4m
n=105: [ 0, 12,p] (;1,2,4,8,16,32,-41,23,46,-13,-26,-52)p
n=106: [ 1, 26,m]
(1;2,4,8,16,32,-42,22,44,-18,-36,34,-38,30,-46,14,28,-50,6,12,
24,48,-10,-20,-40,26,52)m
n=107: [ 0, 53,m]*
(;1,2,4,8,16,32,-43,21,42,-23,-46,15,30,-47,13,26,52,-3,-6,
-12,-24,-48,11,22,44,-19,-38,31,-45,17,34,-39,29,-49,9,18
,
36,-35,37,-33,41,-25,-50,7,14,28,-51,5,10,20,40,-27,53)m
n=108: [ 2, 9,m] (1,2;4,8,16,32,-44,20,40,-28,52)m
n=109: [ 0, 18,m]
(;1,2,4,8,16,32,-45,19,38,-33,43,-23,-46,17,34,-41,27,54)m
n=110: [ 1, 20,p]
(1;2,4,8,16,32,-46,18,36,-38,34,-42,26,52,-6,-12,-24,-48,14,
28,-54)p
n=111: [ 0, 36,p]
(;1,2,4,8,16,32,-47,17,34,-43,25,50,-11,-22,-44,23,46,-19,
-38,35,-41,29,-53,5,10,20,40,-31,49,-13,-26,-52,7,14,28,-
55)p
n=112: [ 4, 3,p] (1,2,4,8;16,32,-4p
n=113: [ 0, 14,m] (;1,2,4,8,16,32,-49,15,30,-53,7,14,28,56)m
n=114: [ 1, 9,m] (1;2,4,8,16,32,-50,14,28,56)m
n=115: [ 0, 44,p]
(;1,2,4,8,16,32,-51,13,26,52,-11,-22,-44,27,54,-7,-14,-28,
-56,3,6,12,24,48,-19,-38,39,-37,41,-33,49,-17,-34,47,-21,
-42,31,-53,9,18,36,-43,29,-57)p
n=116: [ 2, 14,m] (1,2;4,8,16,32,-52,12,24,48,-20,-40,36,-44,28,56)m
n=117: [ 0, 12,p] (;1,2,4,8,16,32,-53,11,22,44,-29,-5p
n=118: [ 1, 29,m] (1;2,4,8,16,32,-54,10,20,40,-38,42,-34,50,-18,-36,46,-26,
-52,14,28,56,-6,-12,-24,-48,22,44,-30,5m
n=119: [ 0, 24,p]
(;1,2,4,8,16,32,-55,9,18,36,-47,25,50,-19,-38,43,-33,53,-13,
-26,-52,15,30,-59)p
n=120: [ 3, 4,p] (1,2,4;8,16,32,-56)p
n=121: [ 0, 55,m]
(;1,2,4,8,16,32,-57,7,14,28,56,-9,-18,-36,49,-23,-46,29,58,
-5,-10,-20,-40,41,-39,43,-35,51,-19,-38,45,-31,59,-3,-6,-
12,
-24,-48,25,50,-21,-42,37,-47,27,54,-13,-26,-52,17,34,-53,
15,
30,60)m
n=122: [ 1, 30,m]
(1;2,4,8,16,32,-58,6,12,24,48,-26,-52,18,36,-50,22,44,-34,
54,-14,-28,-56,10,20,40,-42,38,-46,30,60)m
n=123: [ 0, 20,p]
(;1,2,4,8,16,32,-59,5,10,20,40,-43,37,-49,25,50,-23,-46,31,
-61)p
n=124: [ 2, 5,p] (1,2;4,8,16,32,-60)p
n=125: [ 0, 50,m]
(;1,2,4,8,16,32,-61,3,6,12,24,48,-29,-58,9,18,36,-53,19,38,
-49,27,54,-17,-34,57,-11,-22,-44,37,-51,23,46,-33,59,-7,
-14,-28,-56,13,26,52,-21,-42,41,-43,39,-47,31,62)m
n=126: [ 1, 6,p] (1;2,4,8,16,32,-62)p
n=127: [ 0, 7,p] (;1,2,4,8,16,32,-63)p
n=128: [ 7, 1,o] (1,2,4,8,16,32,64;0)o
n=129: [ 0, 7,m] (;1,2,4,8,16,32,64)m
n=130: [ 1, 6,m] (1;2,4,8,16,32,64)m
n=131: [ 0, 65,m]*
(;1,2,4,8,16,32,64,-3,-6,-12,-24,-48,35,-61,9,18,36,-59,13,

26,52,-27,-54,23,46,-39,53,-25,-50,31,62,-7,-14,-28,-56,19,

38,-55,21,42,-47,37,-57,17,34,-63,5,10,20,40,-51,29,58,-15,
-30,-60,11,22,44,-43,45,-41,49,-33,65)m
n=132: [ 2, 5,m] (1,2;4,8,16,32,64)m
n=133: [ 0, 18,p]
(;1,2,4,8,16,32,64,-5,-10,-20,-40,53,-27,-54,25,50,-33,-66)p
n=134: [ 1, 33,m]
(1;2,4,8,16,32,64,-6,-12,-24,-48,38,-58,18,36,-62,10,20,40,
-54,26,52,-30,-60,14,28,56,-22,-44,46,-42,50,-34,66)m
n=135: [ 0, 36,p]
(;1,2,4,8,16,32,64,-7,-14,-28,-56,23,46,-43,49,-37,61,-13,
-26,-52,31,62,-11,-22,-44,47,-41,53,-29,-58,19,38,-59,17,
34,-67)p
n=136: [ 3, 4,m] (1,2,4;8,16,32,64)m
n=137: [ 0, 34,m]
(;1,2,4,8,16,32,64,-9,-18,-36,65,-7,-14,-28,-56,25,50,-37,

63,-11,-22,-44,49,-39,59,-19,-38,61,-15,-30,-60,17,34,6m
n=138: [ 1, 22,p]
(1;2,4,8,16,32,64,-10,-20,-40,58,-22,-44,50,-38,62,-14,-28,
-56,26,52,-34,-6p
n=139: [ 0, 69,m]*
(;1,2,4,8,16,32,64,-11,-22,-44,51,-37,65,-9,-18,-36,67,-5,
-10,-20,-40,59,-21,-42,55,-29,-58,23,46,-47,45,-49,41,-57
,

25,50,-39,61,-17,-34,-68,3,6,12,24,48,-43,53,-33,-66,7,14,

28,56,-27,-54,31,62,-15,-30,-60,19,38,-63,13,26,52,-35,69)m
n=140: [ 2, 12,p] (1,2;4,8,16,32,64,-12,-24,-48,44,-52,36,-6p
n=141: [ 0, 46,p] (;1,2,4,8,16,32,64,-13,-26,-52,37,-67,7,14,28,56,-29,-58,

25,50,-41,59,-23,-46,49,-43,55,-31,-62,17,34,68,-5,-10,-20,
-40,61,-19,-38,65,-11,-22,-44,53,-35,-70)p
n=142: [ 1, 35,p] (1;2,4,8,16,32,64,-14,-28,-56,30,60,-22,-44,54,-34,-68,6,
12,24,48,-46,50,-42,58,-26,-52,38,-66,10,20,40,-62,18,36,
-70)p
n=143: [ 0, 60,p]
(;1,2,4,8,16,32,64,-15,-30,-60,23,46,-51,41,-61,21,42,-59,

25,50,-43,57,-29,-58,27,54,-35,-70,3,6,12,24,48,-47,49,-45,

53,-37,69,-5,-10,-20,-40,63,-17,-34,-68,7,14,28,56,-31,-62,
19,38,-67,9,18,36,-71)p
n=144: [ 4, 3,m] (1,2,4,8;16,32,64)m
n=145: [ 0, 14,m] (;1,2,4,8,16,32,64,-17,-34,-68,9,18,36,72)m
n=146: [ 1, 9,p] (1;2,4,8,16,32,64,-18,-36,-72)p
n=147: [ 0, 42,p] (;1,2,4,8,16,32,64,-19,-38,71,-5,-10,-20,-40,67,-13,-26,
-52,43,-61,25,50,-47,53,-41,65,-17,-34,-68,11,22,44,-59,2
9,
58,-31,-62,23,46,-55,37,-73)p
n=148: [ 2, 18,m]
(1,2;4,8,16,32,64,-20,-40,68,-12,-24,-48,52,-44,60,-28,-56,
36,72)m
n=149: [ 0, 74,m]* (;1,2,4,8,16,32,64,-21,-42,65,-19,-38,73,-3,-6,-12,-24,
-48,53,-43,63,-23,-46,57,-35,-70,9,18,36,72,-5,-10,-20,
-40,69,-11,-22,-44,61,-27,-54,41,-67,15,30,60,-29,-58,33,
66,-17,-34,-68,13,26,52,-45,59,-31,-62,25,50,-49,51,-47,
55,-39,71,-7,-14,-28,-56,37,74)m
n=150: [ 1, 20,p]
(1;2,4,8,16,32,64,-22,-44,62,-26,-52,46,-58,34,68,-14,-28,
-56,38,-74)p
n=151: [ 0, 15,p] (;1,2,4,8,16,32,64,-23,-46,59,-33,-66,19,38,-75)p
n=152: [ 3, 9,m] (1,2,4;8,16,32,64,-24,-48,56,-40,72)m
n=153: [ 0, 24,p]
(;1,2,4,8,16,32,64,-25,-50,53,-47,59,-35,-70,13,26,52,-49,
55,-43,67,-19,-38,-76)p
n=154: [ 1, 30,p] (1;2,4,8,16,32,64,-26,-52,50,-54,46,-62,30,60,-34,-68,18,
36,72,-10,-20,-40,74,-6,-12,-24,-48,58,-38,-76)p
n=155: [ 0, 20,p] (;1,2,4,8,16,32,64,-27,-54,47,-61,33,66,-23,-46,63,-29,
-58,39,-77)p
n=156: [ 2, 12,p] (1,2;4,8,16,32,64,-28,-56,44,-68,20,40,-76)p
n=157: [ 0, 26,m] (;1,2,4,8,16,32,64,-29,-58,41,-75,7,14,28,56,-45,67,-23,
-46,65,-27,-54,49,-59,39,7m
n=158: [ 1, 39,p]
(1;2,4,8,16,32,64,-30,-60,38,76,-6,-12,-24,-48,62,-34,-68,

22,44,-70,18,36,72,-14,-28,-56,46,-66,26,52,-54,50,-58,42,
-74,10,20,40,-7p
n=159: [ 0, 52,p] (;1,2,4,8,16,32,64,-31,-62,35,70,-19,-38,-76,7,14,28,56,
-47,65,-29,-58,43,-73,13,26,52,-55,49,-61,37,74,-11,-22,

-44,71,-17,-34,-68,23,46,-67,25,50,-59,41,-77,5,10,20,40,
-79)p
n=160: [ 5, 2,m] (1,2,4,8,16;32,64)m
n=161: [ 0, 33,p] (;1,2,4,8,16,32,64,-33,-66,29,58,-45,71,-19,-38,-76,9,18,
36,72,-17,-34,-68,25,50,-61,39,78,-5,-10,-20,-40,-80)p
n=162: [ 1, 27,m] (1;2,4,8,16,32,64,-34,-68,26,52,-58,46,-70,22,44,-74,14,
28,56,-50,62,-38,-76,10,20,40,80)m
n=163: [ 0, 81,m]* (;1,2,4,8,16,32,64,-35,-70,23,46,-71,21,42,-79,5,10,20,
40,80,-3,-6,-12,-24,-48,67,-29,-58,47,-69,25,50,-63,37,

74,-15,-30,-60,43,-77,9,18,36,72,-19,-38,-76,11,22,44,-75,

13,26,52,-59,45,-73,17,34,68,-27,-54,55,-53,57,-49,65,-33,
-66,31,62,-39,-78,7,14,28,56,-51,61,-41,81)m
n=164: [ 2, 10,m] (1,2;4,8,16,32,64,-36,-72,20,40,80)m
n=165: [ 0, 20,p] (;1,2,4,8,16,32,64,-37,-74,17,34,68,-29,-58,49,-67,31,62,
-41,-82)p
n=166: [ 1, 41,m] (1;2,4,8,16,32,64,-38,-76,14,28,56,-54,58,-50,66,-34,-68,

30,60,-46,74,-18,-36,-72,22,44,-78,10,20,40,80,-6,-12,-24,
-48,70,-26,-52,62,-42,82)m
n=167: [ 0, 83,p] (;1,2,4,8,16,32,64,-39,-78,11,22,44,-79,9,18,36,72,-23,
-46,75,-17,-34,-68,31,62,-43,81,-5,-10,-20,-40,-80,7,14,
28,56,-55,57,-53,61,-45,77,-13,-26,-52,63,-41,-82,3,6,12,
24,48,-71,25,50,-67,33,66,-35,-70,27,54,-59,49,-69,29,58,
-51,65,-37,-74,19,38,76,-15,-30,-60,47,-73,21,42,-83)p
n=168: [ 3, 6,p] (1,2,4;8,16,32,64,-40,-80)p
n=169: [ 0, 78,m] (;1,2,4,8,16,32,64,-41,-82,5,10,20,40,80,-9,-18,-36,-72,
25,50,-69,31,62,-45,79,-11,-22,-44,81,-7,-14,-28,-56,57,
-55,59,-51,67,-35,-70,29,58,-53,63,-43,83,-3,-6,-12,-24,
-48,73,-23,-46,77,-15,-30,-60,49,-71,27,54,-61,47,-75,19,
38,76,-17,-34,-68,33,66,-37,-74,21,42,84)m
n=170: [ 1, 8,p] (1;2,4,8,16,32,64,-42,-84)p
n=171: [ 0, 9,m] (;1,2,4,8,16,32,64,-43,85)m
n=172: [ 2, 7,m] (1,2;4,8,16,32,64,-44,84)m
n=173: [ 0, 86,m]* (;1,2,4,8,16,32,64,-45,83,-7,-14,-28,-56,61,-51,71,-31,
-62,49,-75,23,46,-81,11,22,44,-85,3,6,12,24,48,-77,19,38,

76,-21,-42,-84,5,10,20,40,80,-13,-26,-52,69,-35,-70,33,66,
-41,-82,9,18,36,72,-29,-58,57,-59,55,-63,47,-79,15,30,60,
-53,67,-39,-78,17,34,68,-37,-74,25,50,-73,27,54,-65,43,86
)m
n=174: [ 1, 28,p] (1;2,4,8,16,32,64,-46,82,-10,-20,-40,-80,14,28,56,-62,50,
-74,26,52,-70,34,68,-38,-76,22,44,-86)p
n=175: [ 0, 60,p] (;1,2,4,8,16,32,64,-47,81,-13,-26,-52,71,-33,-66,43,86,
-3,-6,-12,-24,-48,79,-17,-34,-68,39,78,-19,-38,-76,23,46,
-83,9,18,36,72,-31,-62,51,-73,29,58,-59,57,-61,53,-69,37,
74,-27,-54,67,-41,-82,11,22,44,-87)p
n=176: [ 4, 5,m] (1,2,4,8;16,32,64,-48,80)m
n=177: [ 0, 29,m] (;1,2,4,8,16,32,64,-49,79,-19,-38,-76,25,50,-77,23,46,
-85,7,14,28,56,-65,47,-83,11,22,44,8m
n=178: [ 1, 11,p] (1;2,4,8,16,32,64,-50,78,-22,-44,-8p
n=179: [ 0, 89,m]*
(;1,2,4,8,16,32,64,-51,77,-25,-50,79,-21,-42,-84,11,22,44,
88,-3,-6,-12,-24,-48,83,-13,-26,-52,75,-29,-58,63,-53,73,
-33,-66,47,-85,9,18,36,72,-35,-70,39,78,-23,-46,87,-5,-10
,
-20,-40,-80,19,38,76,-27,-54,71,-37,-74,31,62,-55,69,-41,
-82,15,30,60,-59,61,-57,65,-49,81,-17,-34,-68,43,86,-7,-1
4,
-28,-56,67,-45,89)m
n=180: [ 2, 12,p] (1,2;4,8,16,32,64,-52,76,-28,-56,68,-44,-8p
n=181: [ 0, 90,m]* (;1,2,4,8,16,32,64,-53,75,-31,-62,57,-67,47,-87,7,14,28,

56,-69,43,86,-9,-18,-36,-72,37,74,-33,-66,49,-83,15,30,60,
-61,59,-63,55,-71,39,78,-25,-50,81,-19,-38,-76,29,58,-65,
51,-79,23,46,-89,3,6,12,24,48,-85,11,22,44,88,-5,-10,-20,
-40,-80,21,42,84,-13,-26,-52,77,-27,-54,73,-35,-70,41,82,
-17,-34,-68,45,90)m
n=182: [ 1, 12,p] (1;2,4,8,16,32,64,-54,74,-34,-68,46,-90)p
n=183: [ 0, 60,p] (;1,2,4,8,16,32,64,-55,73,-37,-74,35,70,-43,-86,11,22,44,
88,-7,-14,-28,-56,71,-41,-82,19,38,76,-31,-62,59,-65,53,
-77,29,58,-67,49,-85,13,26,52,-79,25,50,-83,17,34,68,-47,
89,-5,-10,-20,-40,-80,23,46,-91)p
n=184: [ 3, 11,p] (1,2,4;8,16,32,64,-56,72,-40,-80,24,48,-8p
n=185: [ 0, 18,m] (;1,2,4,8,16,32,64,-57,71,-43,-86,13,26,52,-81,23,46,92)m
n=186: [ 1, 10,p] (1;2,4,8,16,32,64,-58,70,-46,-92)p
n=187: [ 0, 40,p]
(;1,2,4,8,16,32,64,-59,69,-49,89,-9,-18,-36,-72,43,86,-15,
-30,-60,67,-53,81,-25,-50,87,-13,-26,-52,83,-21,-42,-84,
19,38,76,-35,-70,47,-93)p
n=188: [ 2, 23,p] (1,2;4,8,16,32,64,-60,68,-52,84,-20,-40,-80,28,56,-76,36,
72,-44,-88,12,24,48,-92)p
n=189: [ 0, 18,p] (;1,2,4,8,16,32,64,-61,67,-55,79,-31,-62,65,-59,71,-47,
-94)p
n=190: [ 1, 36,p] (1;2,4,8,16,32,64,-62,66,-58,74,-42,-84,22,44,88,-14,-28,
-56,78,-34,-68,54,-82,26,52,-86,18,36,72,-46,-92,6,12,24,
48,-94)p
n=191: [ 0, 95,p] (;1,2,4,8,16,32,64,-63,65,-61,69,-53,85,-21,-42,-84,23,
46,92,-7,-14,-28,-56,79,-33,-66,59,-73,45,90,-11,-22,-44,
-88,15,30,60,-71,49,-93,5,10,20,40,80,-31,-62,67,-57,77,
-37,-74,43,86,-19,-38,-76,39,78,-35,-70,51,-89,13,26,52,
-87,17,34,68,-55,81,-29,-58,75,-41,-82,27,54,-83,25,50,
-91,9,18,36,72,-47,-94,3,6,12,24,48,-95)p
n=192: [ 6, 1,m] (1,2,4,8,16,32;64)m
n=193: [ 0, 48,m]
(;1,2,4,8,16,32,64,-65,63,-67,59,-75,43,86,-21,-42,-84,25,

50,-93,7,14,28,56,-81,31,62,-69,55,-83,27,54,-85,23,46,92,
-9,-18,-36,-72,49,-95,3,6,12,24,48,96)m
n=194: [ 1, 24,m] (1;2,4,8,16,32,64,-66,62,-70,54,-86,22,44,88,-18,-36,-72,
50,-94,6,12,24,48,96)m
n=195: [ 0, 12,p] (;1,2,4,8,16,32,64,-67,61,-73,49,-97)p
n=196: [ 2, 21,p] (1,2;4,8,16,32,64,-68,60,-76,44,88,-20,-40,-80,36,72,-52,
92,-12,-24,-48,-96)p
n=197: [ 0, 98,m]* (;1,2,4,8,16,32,64,-69,59,-79,39,78,-41,-82,33,66,-65,67,
-63,71,-55,87,-23,-46,-92,13,26,52,-93,11,22,44,88,-21,-4
2,
-84,29,58,-81,35,70,-57,83,-31,-62,73,-51,95,-7,-14,-28,
-56,85,-27,-54,89,-19,-38,-76,45,90,-17,-34,-68,61,-75,47
,

94,-9,-18,-36,-72,53,-91,15,30,60,-77,43,86,-25,-50,97,-3,
-6,-12,-24,-48,-96,5,10,20,40,80,-37,-74,49,9m
n=198: [ 1, 15,m] (1;2,4,8,16,32,64,-70,58,-82,34,68,-62,74,-50,9m
n=199: [ 0, 99,p] (;1,2,4,8,16,32,64,-71,57,-85,29,58,-83,33,66,-67,65,-69,
61,-77,45,90,-19,-38,-76,47,94,-11,-22,-44,-88,23,46,92,
-15,-30,-60,79,-41,-82,35,70,-59,81,-37,-74,51,-97,5,10,
20,40,80,-39,-78,43,86,-27,-54,91,-17,-34,-68,63,-73,53,
-93,13,26,52,-95,9,18,36,72,-55,89,-21,-42,-84,31,62,-75,
49,98,-3,-6,-12,-24,-48,-96,7,14,28,56,-87,25,50,-99)p
n=200: [ 3, 10,m] (1,2,4;8,16,32,64,-72,56,-88,24,48,96)m

With kind regards,
Rainer Rosenthal
r.rosenthal@web.de
Ari
science forum beginner

Joined: 24 Mar 2005
Posts: 30

Posted: Fri Mar 25, 2005 5:27 pm    Post subject: Re: Given a plane convex closed curve

"Valeri Astanoff" <astanoff@yahoo.fr> wrote in message news:999ce89a.0503240132.1528c00d@posting.google.com...
 Quote: Hi group, Given a plane convex closed curve, show that its ratio diameter/length is at most pi. I'm unable to find a rigorous proof. Thanks in advance for any help. Valeri.

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Maybe the calculus of variations is needed.
Am not sure about the details

Aristotle Polonium

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
John Gabbriel
science forum beginner

Joined: 29 Apr 2005
Posts: 23

Posted: Sat Mar 26, 2005 5:37 am    Post subject: Re: Given a plane convex closed curve

astanoff@yahoo.fr wrote:
 Quote: Thank you for your explanation. By the way, can the curve you use be called an "offset" curve? v.a.

Never heard that term before.
Peter Webb
science forum Guru Wannabe

Joined: 05 May 2005
Posts: 192

Posted: Fri Apr 29, 2005 11:35 am    Post subject: Re: Because "I are one"

"Tim Little" <tim-from-google@little-possums.net> wrote in message
 Quote: "Peter Webb" wrote: What is the "indirect evidence" of wavlengths greater than 10^12 metres? "Variations in magnetospheric phenomena" -- unfortunately I don't know to which magnetosphere it is referring, or what sort of phenomena. Secondly, nobody seriously believes that Maxwells equations don't hold true at all frequencies To the contrary! Maxwell's equations almost certainly don't hold at universal scales, because spacetime is so warped on that scale that Maxwell's equations are not even a useful approximation. Furthermore, the entire observable universe would be in the near field, and hence it would not really qualify as radiation.

 Quote: On the other end, there is probably an lower limit to wavelength again set by GR, where unknown quantum gravity effects are likely to take over. I can understand practical issues at high frequencies - because the photon may decay into two or more particles No isolated massless particle can decay. Indeed, photons have been observed with energies billions of times greater than the energy required to decay into particles, if it were possible.

I stand corrected. Perhaps you can explain something for me. I understand
that a particle/anti-particle annihalation can spawn TWO photons moving in
opposite directions, and this reversable (?). However, for the purposes of
conserving energy and momentum (etc) precisely, the two photons would need
precisely the correct energy - no error whatsoever allowed. The only way I
can think these photons could be prepared is the same or different
particle/anti-particle annihalation. Is this true? Or can you artificially
tune photons (through some Gamma wave Laser thingie) that can pair up to
produce particle anti-particle creations?

 Quote: I can directly detect a radio wave of frequency 6 x 10^6 metres, without moving from my chair, by my moving my speakers next to my monitor and seeing the 50 Hz (I'm an Australian) RF interference. That's not direct detection of electromagnetic radiation, since in that case your monitor is certainly in the near field of the speakers. It's almost certainly responding directly to a dominant magnetic field, not to a propagating electromagnetic wave where the electric and magnetic components of the field have equal energy.

Who said anything about having to detect EM radiation through the E
component. That's E-centricity at its worst.

And where there is an M changing there is an E.

And whether this is the near wave or the propogating wave, its evidence that
Maxwell works down to 50 Hz.

 Quote: 50 Hz radio waves do exist, but moving your speakers next to your monitor is not evidence of their existence. - Tim
Peter Webb
science forum Guru Wannabe

Joined: 05 May 2005
Posts: 192

Posted: Fri Apr 29, 2005 11:38 am    Post subject: Re: Joke Schema

"William Elliot" <marsh@hevanet.remove.com> wrote in message
news:Pine.BSI.4.58.0504282145330.14465@vista.hevanet.com...
 Quote: On Fri, 29 Apr 2005, Mike Terry wrote: For example: What's A, B, and Cs Ds? - - - - - - - - - spoiler... - - Whoops, it took so long to call my page then the off duty page and then wake the sleeping page and then call back the page on vacation, that the answer spoiled before it was it could be served.

Yeah, timing's everything in typing a joke.
MJ
science forum beginner

Joined: 29 Apr 2005
Posts: 32

Posted: Fri Apr 29, 2005 1:01 pm    Post subject: Re: The null joke

"Bob Pease" <pope@youkno.net> wrote in message
news:d4se0d\$glj@dispatch.concentric.net...

 Quote: No. It's a prequisite for Buddhism 101. In Buddhism 501, they do nothing, and with complete and non-judgmental awareness in every moment, realizing the ultimate emptiness of it all (at the absolute level). Mj

 Quote: In Buddhism 502 they get drunk, and visit Buddhism 501 classes, giggle, and let a lot of farts. RJ P Right on, and in Buddhism 503, they all get back to ordinary life. MJ

 Quote: It's like the used car salesman who became a guru. He went back to the Used car business when someone asked him "You say ..."We are all ONE!!!" ... Well, I wanna know, we are all one... WHAT???" Shantih RJ P

If *he* didn't know the answer, who does? MJ
MJ
science forum beginner

Joined: 29 Apr 2005
Posts: 32

Posted: Fri Apr 29, 2005 1:18 pm    Post subject: Re: The null joke

"William Elliot" <marsh@hevanet.remove.com> wrote in message
news:Pine.BSI.4.58.0504282154400.14465@vista.hevanet.com...

 Quote: No. It's a prequisite for Buddhism 101. In Buddhism 501, they do nothing, and with complete and non-judgmental awareness in every moment, realizing the ultimate emptiness of it all (at the absolute level). In Buddhism 502 they get drunk, and visit Buddhism 501 classes, giggle, and let a lot of farts. RJ P Right on, and in Buddhism 503, they all get back to ordinary life. Buddhism 501, 502, 503 is too much no empty mind.

In Buddhism 503, one learns to live ordinary life with mind empty
of discursive thinking, hesitation, doubt and confusion,
preferably experientally.

 Quote: "The Tao which is the true Tao cannot be expressed in words."

Excluding the above sentence.

 Quote: Oh, that's very clear, if you read further, you've lost the way.

Only if entangled in the duality of "lost" and "found".

 Quote: Buddhism? Of course, there's nothing to it.

Bob Pease
science forum beginner

Joined: 29 Apr 2005
Posts: 47

Posted: Fri Apr 29, 2005 5:50 pm    Post subject: Re: The null joke

"MJ" <MJ@mj> wrote in message news:a-GdnTyFkLPQ0e_fRVn-rQ@comcast.com...
 Quote: "Bob Pease" wrote in message news:d4se0d\$glj@dispatch.concentric.net... No. It's a prequisite for Buddhism 101. In Buddhism 501, they do nothing, and with complete and non-judgmental awareness in every moment, realizing the ultimate emptiness of it all (at the absolute level). Mj In Buddhism 502 they get drunk, and visit Buddhism 501 classes, giggle, and let a lot of farts. RJ P Right on, and in Buddhism 503, they all get back to ordinary life. MJ It's like the used car salesman who became a guru. He went back to the Used car business when someone asked him "You say ..."We are all ONE!!!" ... Well, I wanna know, we are all one... WHAT???" Shantih RJ P If *he* didn't know the answer, who does? MJ

"Mu"

Or "Wf"

rj p

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