A279 I have an improper integral question that is bugging me a little bit. I ran the problem through Mathematica, which produced a numerical result, but when I worked the problem out by hand on my chalkboard, it came out divergent. I wil explain. Integrate [ 1 / ( x ( x^2 + 6 ) ) ] dx on the interval [ 1, infinity ) Using partial fractions, I separated this function into the following: Integrate [ (1 / 6x ) - ( x / ( 6 ( x^2 + 6 ) ) ) ] dx on the interval [ 1, infinity ) It was then easy enough to separate these into two integrals and then integrate each one. This results in: ( ( ln |x| ) / 6 ) - ( ( ln |x^2 + 6| ) / 12 ) Now, plugging in for infinity and 1 results in ( infinity + ( ln 7) / 12 ). Obviously, this result is infinity, and is thus divergent. But Mathematica insists that the result is ln (7) / 12. Who is right? And if the machine is better, then what did I do wrong. I have checked my work and have computed the integral correctly.. it only leads me to think that the last step was performed wrong. I did try taking a limit for the first part which came out to infinity - infinity, and when transforming the two fractions into one to use L'Hopital's rule, it came out to infinity / 12, which did not allow for L'Hopital as this is simply infinity. Something is wrong and I cannot figure it out! Any help is greatly appreciated... You can see immediately that it is convergent since: 1 / ( x ( x^2 + 6 ) ) < 1 / x^3 whose integral [1..inf] converges --Lynn How do you plug in for infinity? (log |x|)/6 - (log |x^2 + 6|)/12 = (2(log |x|) - log |x^2 + 6|)/12 = (log(x^2) - log |x^2 + 6|)/12 = (1/12) log(x^2/(x^2 + 6)) value at infinity, that minus the value at 1, which is -(log 7)/12, gives (log 7)/12 as the value of the integral. Let x = sqrt(6) tan(theta) Then the integral becomes (1/6) Int(arcsin(1/sqrt(7)) to pi/2) cos(theta)/sin(theta) d(theta) This integrates to (1/6) ln(sin(theta)) evaluated from arcsin(1/sqrt(7)) to pi/2 which gives (1/12) ln(7) Bob factorise 9889. I did it in 500 metres walking., Danny Kodicek Who declares the correct spelling of a word? If three dictionaries spell a word one way and a fourth spells it another way, which is the right one? In your case I guess it is the one that you like the most. There is no right or wrong way to spell something, there is an commonly accepted way though. Whomever invented the word of course. Dimbulb! I deleted the blog mathforprofit a couple of months ago, and someone has taken it over, and now posts as me. Yup, a weird variation on identity theft, and I told Google but they blew me off, saying they don't manage content. So, once again, math people show just how strange they can be, and I learn a lesson that I can't just delete off blogs! You people are rather odd. longer want to use that blog, as math people are weird. James Harris Hey, how do we know you're the original James Harris and not this imposter (if, indeed, there is an imposter?) In that spirit, I'll continue with: Google Group's guide to Net ettiquette says: Something everyone should keep in mind. --- Christopher Heckman (the real one) X-RFC2646: Format=Flowed; Original [JSH] This was well-known, and discussed, on sci.math some weeks ago. Some people are. It was almost universally condemned on sci.math, because setting up such a site is so obviously so sleazy a thing to do to you. It's despicable, IMO. That you sometimes do despicable things online too doesn't excuse it. Nobody on sci.math owned up to doing it, or to knowing who did it. If the perpetrator's identity does become known, you won't be the only one condemning them for it. Now that you can't delete stuff has the feeling of power been replaced by a feeling of impotence? How do you know it was a math person? A more likely perp would be someone from one of those numerous newsgroups you post off-topic material to, like sci.physics or sci.crypt. You math amateurs are really stupid. Are you sure? Rich Yup, personal experience. This seems like a classic question and so, though quite belatedly, I looked for a nice closed form approximation. My result is a Pade approximation. The crux of the problem, at least as I approached it, is to invert the function f(a) = a (cosh(x/a) - 1). Crudely, an approximate solution of a (cosh(x/a) - 1) = c is a = x^2/(2c). This simple approximation can be improved substantially by multiplying it by even rational functions, homogeneous in x and c. For example, (Please view in a fixed-width font.) 2 4 2 2 4 x (29295 x + 53025 c x + 17032 c ) a = ----------------------------------- (*) 4 2 2 4 2 c (29295 x + 43260 c x + 7820 c ) To illustrate using this approximation, we consider the specific problem posed by Sam, for which we have 30 = a cosh(0/a) + b and 80 = a cosh(140/a) + b. Subtracting the first from the second equation, we get 50 = a (cosh(140/a) - 1). Then using (*), putting x = 140 and c = 50, we have approximately a = 13679469356/67112911 = 203.827686... [Compare that with Lynn's a = 203.8277614 .] Finally, b = 30 - a = -173.827686... approximately. Note that approximation (*) does not work well when c/x is large, in which case a different sort of approximation (perhaps using logarithms or the Lambert W function) would be appropriate. David W. Cantrell don.mcdonald 27.02.05 23:23 23:49 SENT. FWD:42nd Mersenne Prime Found : 42nd Mersenne Prime Found : It's 2^ 25,964,951 - 1, and it has 7,816,230 digits. Still far below the : 10 million digits required for the 100,000$. There's still hope! ... summary primes. 25964699 25964717 25964737 25964749 25964761 25964789 25964843 25964857 25964867 25964879 25964891 25964909 25964921 25964927 25964929 **25964951*** 42nd known mersenne prime exponent 27.02.05 17:29 exponent. 25964957** 25964959**twin prime 25964971 25964977 25965007 25965029 25965031 25965041 25965053 25965089 25965097 25965101 ... twin primes? 25962967+4. 25962971+2. 25962973+4. 25963039+4. 25963109+2. 25963121+2. 25963241+2. 25963723+4. 25963739+2. 25963789+4. 25964231+2. 25964381+2. 25964383+4. 25964569+4. 25964927+2.** 25964957+2. 42nd mersenne prime exponent = 259 649 51. ********25 million. Febr. 2005. not a twin prime. 25965029+2. 25965097+4. 25965223+4. 25965299+2. 25965347+2. 25965427+4. 25965437+2. 25965517+4. ... moddat.? factors. next prime 25964921 ,,, (2k=6p+1, 10LOG=81) 155789527 |divides 2^ 25964921 -1 .. csec=17 divisors of n-1 = 25964920 = 1*25964920/= 2*12982460/= 4*6491230/= 5*5192984/= 8*3245615/= 10*2596492/ = 20*1298246/= 40*649123/-1 CONTINUE DON. 18 at erl. 980 PRIME n 25964921 ) n+2 25964922 next prime 25964927 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964927 ) n+2 25964928 next prime 25964929 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964929 ) n+2 25964930 next prime 25964951 ,,,, ********* LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964951 ) n+2 25964952 ** that one actually was 42nd known mersenne prime exponent xxx********* therefore it was one of very few (42 known) that does not have a factor.??; ********* next prime 25964957 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964957 ) n+2 25964958 next prime 25964959 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964959 ) n+2 25964960 next prime 25964971 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964971 ) n+2 25964972 ... next prime 25965143 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25965143 ) n+2 25965144 next prime 25965161 ,,, (2k=6p+1, 10LOG=81) 155790967 |divides 2^ 25965161 -1 .. csec=16 divisors of n-1 = 25965160 .. many divisors. ****** = 1*25965160/= 2*12982580/= 4*6491290/= 5*5193032/= 8*3245645/= 10*2596516/ = 13*1997320/= 20*1298258/= 23*1128920/= 26*998660/= 40*649129/ = 46*564460/= 52*499330/= 65*399464/= 92*282230/= 104*249665/= 115*225784/ = 130*199732/= 167*155480/= 169*153640/= 184*141115/= 230*112892/ = 260*99866/= 299*86840/= 334*77740/= 338*76820/= 460*56446/= 520*49933/ = 598*43420/= 668*38870/= 676*38410/= 835*31096/= 845*30728/= 920*28223/ = 1196*21710/= 1336*19435/= 1352*19205/= 1495*17368/= 1670*15548/ = 1690*15364/= 2171*11960/= 2392*10855/= 2990*8684/= 3340*7774/ = 3380*7682/= 3841*6760/= 3887*6680/= 4342*5980/-1 CONTINUE DON. 18 at erl. 980 PRIME n 25965161 ) n+2 25965162 ... prime2kp???? step 2*b% = 51929902 enter minimum [Return = 3] [N.B.** recommend BASIC 64.**] ? minimum 3 ,, Sint = -1 ***factor = 155,789,707 **** PRIME. (test dup. factors).*** contin cs.= 60 == divides 2^25964951 +1.************** mersenne plus 2.*** , Floating point exception: invalid operation find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 207719609 cs. = 971 at ERL. = 128 BBC Basic 64 Prog $..calc.normal.IsPriMe2kp, e n d . ... profilac.?? NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964945,6.741783688E14, 5095.58 5.5192989 3.851346498 (1) 25928464 +36481 = 5092^2 + 191^2. (2) 15673681 +10291264 = 3959^2 + 3208^2. 25964946,6.741784207E14, 5095.58 2.3.3.7.251.821 3.85134635 25964947,6.741784726E14, 5095.58 29.895343 3.851346202 25964948,6.741785245E14, 5095.58 2.2.43.150959 3.851346053 25964949,6.741785767E14, 5095.58 3.31.89.3137 3.851345905 NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964950,6.741786286E14, 5095.58 2.5.5.11.17.2777 3.851345756 25964951,6.741786805E14, 5095.58 25964951 *** 3.851345608 42nd mersenne prime exponent. ******** 25964952,6.741787324E14, 5095.58 2.2.2.3.13.83221 3.851345459 25964953,6.741787843E14, 5095.58 7.7.23.23039 3.851345311 ... primecabi ?? ** find (first)factors p, satisfying , S% = m 25964951 ABS( 1.000000000 * 2.000000000 ^ 25964951.000000000 + 0.000000000 ) <= 35.000000000 MOD p. highest factor tested, P% = 6569.000000000 cs. = 5223.000000000 at ERL. = 1430.000000000 summary factors, enter cont.? sumry factors, S% = 0 2 cont. sumry factors, S% = -1 3 cont. (MERSENNE here, S% =+1.) sumry factors, S% = -2 5 S% = +2 11 cont. sumry factors, S% = -3 7 cont. sumry factors, S% = -4 9 S% = +4 5519 cont. sumry factors, S% = -5 0 S% = +5 27 cont. sumry factors, S% = -6 13 cont. sumry factors, S% = -7 15 S% = +7 67 cont. sumry factors, S% = -8 17 S% = +8 173 cont. , sumry factors, S% = -10 21 S% = +10 2609 cont. sumry factors, S% = -11 43 cont. sumry factors, S% = -12 0 S% = +12 47 cont. sumry factors, S% = -13 149 cont. sumry factors, S% = -14 1723 S% = +14 419 cont. sumry factors, S% = -15 0 S% = +15 277 cont. sumry factors, S% = -16 59 S% = +16 233 cont. sumry factors, S% = -17 49 S% = +17 271 cont. sumry factors, S% = -18 0 S% = +18 1171 cont. sumry factors, S% = -19 39 S% = +19 71 cont. sumry factors, S% = -20 0 S% = +20 677 cont. sumry factors, S% = -21 701 cont. sumry factors, S% = -22 45 cont. sumry factors, S% = -23 523 S% = +23 367 cont. sumry factors, S% = -24 139 cont. sumry factors, S% = -25 323 cont. sumry factors, S% = -26 53 S% = +26 493 cont. sumry factors, S% = -27 0 S% = +27 167 cont. , sumry factors, S% = -29 479 S% = +29 131 cont. , sumry factors, S% = -31 0 S% = +31 6473 cont. sumry factors, S% = -32 109 S% = +32 73 cont. sumry factors, S% = -33 107 cont. sumry factors, S% = -34 3499 cont. , BBC Basic V Program $.donmcd.calc.normal.PrimeCabi, e n d . 17/6/94, 5.01.04. ... details. and working. [ LONG 1170 lines. ]********** don.mcdonald. February 26, 2005--Less than a year after the 41st Mersenne prime was reported ( MathWorld Great Internet Mersenne Prime Search (GIMPS) project has discovered the 42nd known Mersenne prime. The candidate prime was flagged prime by an experienced GIMPS volunteer on February 18, independently verified by Tony Reix on Feburary 25, and the exponent was reported on February 26. Mersenne numbers are numbers of the form M n = 2 n - 1, giving the first few as 1, 3, 7, 15, 31, 63, 127, .... Interestingly, the definition of these numbers therefore means that the n th Mersenne number is simply a string of n 1s when represented in binary . For example, M 7= 27- 1 = 127 = 11111112is ???xx a Mersenne number. Mersenne primes are Mersenne numbers that are also prime , i. ... Go to Google Groups Home Groups Results 1 - 1 of 1 for 25964951 . ( 0.21 seconds) Sorted by relevance Sort by date Related groups: utexas.class.cs378-downing 42nd Mersenne Prime Found It's 2^ 25,964,951 - 1, and it has 7,816,230 digits. Still far below the 10 million digits required for the 100,000$. There's still hope! ... utexas.class.cs378-downing - 26 Feb 2005 by Paul Wuersig - View Thread (1 ©2005 Google ... 27.02.05 18:29 yes. Black key Sieve PROC SIEVE PRESS 3 - 6 FOR SIEVE OF (I) PRIMES BBC-B=5, ARCHIMEDES=6, ETC. I% | PRIME | MAX NO.=N% | SIZE=L% PROD(PRIMES)|PROD(PRIME-1) 1 2 2 1 2 3 6 2 3 5 30 8 4 7 210 48 5 11 2310 480 6 13 30030 5760 SIEVE : cont. 1 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 289 293 307 311 313 317 323 331 337 347 349 353 359 361 367 373 379 383 389 391 397 401 409 419 421 431 433 437 439 443 449 457 461 463 467 479 487 491 493 499 503 509 521 523 527 529 541 547 551 557 563 569 571 577 587 589 593 599 601 607 ... ĄTIME = 4.399999999 SECS. PROC CALC CALCULATE PRIMES ABOUT NUMBER ** MAX (30030 ^2 )= 901800900 Q$ = 0 ENTER NUMBER OR EXPRESSION (NEGATIVE = END) ?25964951 mersenne 42nd exponent 25964951 PRESS RET FOR PRIME NOS. SPC FOR FACTORS FROM 25945920 TO 25975950 (PRESS ESCAPE) PROC SIFT / SIFT OUT FACTORS OF SIEVE PROC PRT / PRINTOUT PRIME SEQUENCE 25945937 25945939 25945943 25945957 25945979 25945999 25946003 25946023 25946051 25946059 25946083 25946087 25946099 25946111 25946117 25946119 25946143 25946147 25946149 25946183 25946201 25946237 25946257 25946339 25946357 25946377 25946383 25946407 ... 25964699 25964717 25964737 25964749 25964761 25964789 25964843 25964857 25964867 25964879 25964891 25964909 25964921 25964927 25964929 **25964951*** 42nd known mersenne prime exponent 27.02.05 17:29 exponent. 25964957** 25964959**twin prime 25964971 25964977 25965007 25965029 25965031 25965041 25965053 25965089 25965097 25965101 25965143 25965161 25965209 25965223 25965227 25965257 25965271 25965299 25965301 25965311 25965337 25965347 25965349 25965371 ... 25975553 25975559 25975583 25975627 25975637 25975657 25975661 25975667 25975693 25975723 25975751 25975757 25975759 25975783 25975801 25975811 25975849 25975867 25975877 25975889 25975933 PR%(X%=1743)= 25975933. XTOT% PRIMES SO FAR= 1743 ĄTIME = 2.399999999 SECS. FROM 25975950 TO 26005980 (PRESS ESCAPE) PROC SIFT / SIFT OUT FACTORS OF SIEVE PROC PRT / PRINTOUT PRIME SEQUENCE ... Escape (ERR 17) AT LINE 930 cont. CALCULATE PRIMES ABOUT NUMBER MAX (30030 ^2 )= 901800900 Q$ = 25964951 mersenne 42nd exponent ENTER NUMBER OR EXPRESSION (NEGATIVE = END)?-3 -3 E N D PROG SIEV7/8913 close spool ... FROM 25945920 TO 25975950 (PRESS ESCAPE) PROC SIFT / SIFT OUT FACTORS OF SIEVE PROC PRT / PRINTOUT *twin*PRIME* SEQUENCE 25945937+2. 25945939+4. 25945999+4. 25946083+4. 25946117+2. 25946143+4. 25946147+2. 25946441+2. 25946443+4. 25946579+2. 25946801+2. 25946857+4. 25946861+2. 25946939+2. 25947107+2. 25947167+2. 25947221+2. 25947223+4. 25947239+2. 25947583+4. 25947637+4. 25947641+2. 25947737+2. 25947739+4. 25947829+4. 25947917+2. 25948289+2. 25948337+2. 25948397+2. 25948421+2. ... 25962967+4. 25962971+2. 25962973+4. 25963039+4. 25963109+2. 25963121+2. 25963241+2. 25963723+4. 25963739+2. 25963789+4. 25964231+2. 25964381+2. 25964383+4. 25964569+4. 25964927+2.** 25964957+2. 42nd mersenne prime exponent = 259 649 51. ********25 million. Febr. 2005. not a twin prime. i did not look for triples etc. yet. 25965029+2. 25965097+4. 25965223+4. 25965299+2. 25965347+2. 25965427+4. 25965437+2. 25965517+4. 25965671+2. 25965889+4. 25966091+2. 25966093+4. 25966147+4. 25966159+4. ... 25974911+2. 25974943+4. 25975049+2. 25975123+4. 25975217+2. 25975249+4. 25975307+2. 25975511+2. 25975529+2. 25975549+4. 25975657+4. 25975757+2. PR%(X%=282)= 0. XTOT% PRIMES SO FAR= 282 ĄTIME = 1.43 SECS. FROM 25975950 TO 26005980 (PRESS ESCAPE) PROC SIFT / SIFT OUT FACTORS OF SIEVE PROC PRT / PRINTOUT *twin*PRIME* SEQUENCE Escape ... MAX (30030 ^2 )= 901800900 Q$ = 25964951 mersenne 42 expon ENTER NUMBER OR EXPRESSION (NEGATIVE = END)?-3 -3 E N D PROG SIEV7/twin close *sp. bbc basic v/64 prog calc.factors.LgeFactor.powrabc1, pgms.maths.powabc1, 24.05.98, 5.9.99, 26.11.01, 26.5.02 22-31.8.02 21.2.2005, 42nd mersenne prime (probably) discovered. between 7-10 million digits. mathworld. 2^n -1 shd have difft (fewer?) divs or even be a mersenne prime. ******************* If (n-1) = prime -1 has many divisors, then 2^(n-1)-1 may have many divisors similarly, and 2^n -1 should have different (fewer?) divisors or even be a mersenne prime. But, most of these.. Program powrabc1 has already found a factor (2kp+1) of mersenne numbers (M p = 2^p-1, obviously.) I.e. it eliminates putative mersenne primes! And offers a route to proof of a factor.(?) WHAT IF result? 2^r == 1 mod n for all r | (n-1) ?? THEN 2^(n-1) == 1. AND 2^n == 2 (mod n). Fermat little theorem. This may have a stronger result, I may have heard. yours sincerely Donald.McDonald. n.b. * BASIC64. (menu64) ds mcdonald, wellington, 20/5/99. 14:25 pm. 69,772,567 divides 2^ 11,628,761 -1. 80,812,807 divides 2^13,468,801-1, exponent - 1 has 192 divisors. 39th mersenne prime 24 nov 2001 has 3.5 million digits? spool *ram.moddata from pgms maths powabc1. Sun,27 Feb 2005.17:32:01 enter base [retn 2], a = ? = 2 = Exponents start. reduced printout/ index expression (3.5E6/LOG 2) , return (NEXT = n lower case) 25964000 25964000 25963999 ** ** starting before mersenne exponent.. ** next prime 25964023 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964023 ) n+2 25964024 next prime 25964053 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964053 ) n+2 25964054 next prime 25964063 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964063 ) n+2 25964064 ... next prime 25964371 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964371 ) n+2 25964372 next prime 25964381 ,,, (2k=6p+1, 10LOG=81) 155786287 |divides 2^ 25964381 -1 .. csec=17 divisors of n-1 = 25964380 = 1*25964380/= 2*12982190/= 4*6491095/= 5*5192876/= 10*2596438/ = 13*1997260/= 20*1298219/= 26*998630/= 37*701740/= 52*499315/= 65*399452/ = 74*350870/= 130*199726/= 148*175435/= 185*140348/= 260*99863/ = 370*70174/= 481*53980/= 740*35087/= 962*26990/= 1924*13495/= 2405*10796/ = 2699*9620/= 4810*5398/-1 CONTINUE DON. 18 at erl. 980 PRIME n 25964381 ) n+2 25964382 next prime 25964383 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964383 ) n+2 25964384 ... next prime 25964951 ,,,, ********* LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964951 ) n+2 25964952 ** that one actually was 42nd known mersenne prime exponent xxx********* therefore it was one of very few (42 known) thay does not have a factor.; ********* next prime 25964957 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964957 ) n+2 25964958 next prime 25964959 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964959 ) n+2 25964960 next prime 25964971 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25964971 ) n+2 25964972 ... next prime 25965143 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25965143 ) n+2 25965144 next prime 25965161 ,,, (2k=6p+1, 10LOG=81) 155790967 |divides 2^ 25965161 -1 .. csec=16 divisors of n-1 = 25965160 .. many divisors. ****** = 1*25965160/= 2*12982580/= 4*6491290/= 5*5193032/= 8*3245645/= 10*2596516/ = 13*1997320/= 20*1298258/= 23*1128920/= 26*998660/= 40*649129/ = 46*564460/= 52*499330/= 65*399464/= 92*282230/= 104*249665/= 115*225784/ = 130*199732/= 167*155480/= 169*153640/= 184*141115/= 230*112892/ = 260*99866/= 299*86840/= 334*77740/= 338*76820/= 460*56446/= 520*49933/ = 598*43420/= 668*38870/= 676*38410/= 835*31096/= 845*30728/= 920*28223/ = 1196*21710/= 1336*19435/= 1352*19205/= 1495*17368/= 1670*15548/ = 1690*15364/= 2171*11960/= 2392*10855/= 2990*8684/= 3340*7774/ = 3380*7682/= 3841*6760/= 3887*6680/= 4342*5980/-1 CONTINUE DON. 18 at erl. 980 PRIME n 25965161 ) n+2 25965162 next prime 25965209 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25965209 ) n+2 25965210 ... next prime 25965757 ,,,, LOSS ACCY DON. Z+1. 18 at erl. 1370 PRIME n 25965757 ) n+2 25965758 next prime 25965769 ,, Escape17 at erl. 1290 PRIME n 25965769 ) close #0, 0 = YES/ contin 1 = NEXT ?0 BBC Basic 64 Progm $.donmcd.calc.factors.lgefactor.IsPriMe2kp, 10/3/95, 5.05.98, 26.05.2002 29-8-02. no. plate Mp163 Hanson St, Newt. ** RECOMMENDed BBC BASIC 64. (menu64.) ********************* by Donald S McDonald, 63/3 Hutchison Rd, NEWTOWN, Wellington 2 New Zealand. Phone 64(NZ) +4(WN-) + 389-6820. Solve for (odd) factors, p <= 2^16.5 = 92,681. ABS( C * a^b -i ) <= n MOD p, Progm IsPriMe2kp multiplier * (base ^ index) + offset <= neighbourhood. ARM BBC BASIC VI version 1.05 (C) Acorn 1989 Would you like to see some samples, examples? Yy ( N) ? Example. 10*1E4^7 - 1. Factors of REP.UNIT 29 = 11,111,111, ..., 111. Example. W = 391581*2^216193 -1, Former World's Record largest known Prime No. Guinn Bk of Records 1991. 4W + 3 factors 7* 6199 * 92219 * ... etc. Example. PI = 3 141 592 653 589 793 25-. =3*5*5* 66679* = 131*509* XXX Ex. Bridge, 158 753 389 900. Ex. RUBIK. 43 252 003 274 489 856 000. Ex. Bankcard no., Ex. ATM card no. (Both 16 digits) Ex. 35099 DIVIDES 1011,1213,1415,1617,1819. (20 digits.) Ex. Proves prime up to 2^33-9 = 8E9. BASIC 64 c. 1E14? Ex. 1,234,567,891*1E11 + 011,1213,1415 (21 digits.) Exs. 641 is factor of each following 2^32+1 (Fermat), 5^32+1, 1E32-1 (Rep.unit 32?) 27! + 1 prime? Yes. = 3*(2*11*13)^2*17*19*23 * ( 2^7*3^4*5^2*7 ) ^ 3 +1. contin. Maple V.2 uses a rather weak probabilistic test, try 2,152,302,898,747 = 6763*10627*29947. WelComBBS.SciMath2, 12/3/95. Largest known twin primes.. 1,691,232*1001*1e4020+-1, Dubner, Sci.Math Cole. 2^67-1, search minimum factor = 193.7 million 70*858433+1 DIVIDES 2^858433 -1. (typo CDROM. M 859433 Is prime.) 134MILL DIVIDES M239. 69,728,591 DIVIDES M6972859 13,945,727 DIVIDES 2^6972863-1. M-6972593 ISPRIME JUL.1999. M257 typo. PENG DICT CUR + INTG Nos., Revised edn 1997. = 536006138814359. M 13468801, factor 80812807. 2^74 +1. = 5*149*593*184,481,113*231,769,777. yes (T2100 wordpad.) PROC input. enter text, formula. Analysis of c, a, b, i, n later : ?M-25964951 multiplier * (base ^ index) + offset <= neighbourhood. 2E9 2E9 2E9 2E9 2^5. enter base [2] ( expression a$ , max= 2E9, often 2 , 10, 2^10, 1E3. ) ? a$ = 2 = 2 enter index (power b%, [default 1]) ?25964951 enter multiplier, c% (default = 1) ? 1 enter offset, +/- i% = EVAL i$ (default = -1) ?0 0 = 0 enter neighbourhood (sm. latitude) <= n%, (default = 0 ) ?20 find factors p, satisfying , Sint = M 25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. PROC binary ( 25964951 Hex. 18C3197 bits h = 1 poss. area of improvement 1000^&F ? 2^h = 2 12982475 , 1 6491237 , 1 3245618 , 1 1622809 , 0 811404 , 1 405702 , 0 202851 , 0 101425 , 1 50712 , 1 25356 , 0 12678 , 0 6339 , 0 3169 , 1 1584 , 1 792 , 0 396 , 0 198 , 0 99 , 0 49 , 1 24 , 1 12 , 0 6 , 0 3 , 0 1 , 1 0 , 1 PROC mult. enter test divisors: start$ [Return = 1, .0 = quit] = ? start = 1 test unique even prime (plus powers 2) Spc = Yes, Nn no.? Yes. , ... Sint = 0 factor = 32768 = 2 * 16384 = 4 * 8192 = 8 * 4096 = 16 * 2048 = 32 * 1024 = 64 * 512 = 128 * 256 contin cs.= 4520 contin cs.= 4573 step 2*b% = 51929902 enter minimum [Return = 3] [N.B.** recommend BASIC 64.**] ? minimum 3 ,, Sint = -1 factor = 155,789,707 **** PRIME. (test dup. factors).*** contin cs.= 60 == divides 2^25964951 +1.************** mersenne plus 2. , Floating point exception: invalid operation find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 207719609 cs. = 971 at ERL. = 128 BBC Basic 64 Prog $..calc.normal.IsPriMe2kp, e n d . PROC mult. enter test divisors: start$ [Return = 1, .0 = quit] = ? start = 1 test unique even prime (plus powers 2) Spc = Yes, Nn no.? step 2 = 2 for testing factors of neighbours.** enter minimum [Return = 3] [N.B.** recommend BASIC 64.**] ? minimum 3 , Sint = -1 factor = 3 = 3 * 1PRIME. (test dup. factors). , Sint = -4 factor = 9 = 3 * 3 contin cs.= 13 , Sint = 5 factor = 27 = 3 * 9 contin cs.= 795 , Sint = 5 factor = 81 = 3 * 27 = 9 * 9 contin cs.= 1162 , contin cs.= 1242 , Sint = -2 factor = 5 PRIME. (test dup. factors). , Sint = -2 factor = 25 = 5 * 5 contin cs.= 1330 , Sint = -2 factor = 125 = 5 * 25 contin cs.= 1474 , contin cs.= 1774 , Sint = -3 factor = 7 PRIME. (test dup. factors). , Sint = -17 factor = 49 = 7 * 7 contin cs.= 1859 , contin cs.= 1926 , Sint = -4 factor = 9 = 3 * 3 contin cs.= 2005 , Sint = 2 factor = 11 PRIME. (test dup. factors). , Sint = 2 factor = 121 = 11 * 11 contin cs.= 2088 , contin cs.= 2165 , Sint = -6 factor = 13 PRIME. (test dup. factors). , contin cs.= 2253 , Sint = -7 factor = 15 = 3 * 5 contin cs.= 2324 , Sint = -8 factor = 17 PRIME. (test dup. factors). , contin cs.= 2407 , Sint = -6 factor = 19 PRIME. (test dup. factors). , Sint = -6 factor = 361 = 19 * 19 contin cs.= 2501 , contin cs.= 2526 , Sint = -10 factor = 21 = 3 * 7 contin cs.= 2534 , Sint = 2 factor = 23 PRIME. (test dup. factors). , contin cs.= 2549 , Sint = -2 factor = 25 = 5 * 5 contin cs.= 2556 , Sint = 5 factor = 27 = 3 * 9 contin cs.= 2563 , Sint = -3 factor = 29 PRIME. (test dup. factors). , contin cs.= 2576 , Sint = 2 factor = 31 PRIME. (test dup. factors). , contin cs.= 2589 ... Escape find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 9953 cs. = 14542 at ERL. = 129 BBC Basic 64 Prog $..calc.normal.IsPriMe2kp, e n d . BBC-BASv PROG PROFILAC/, 27.6.91, 7/8/94, 14.06.00, 18.02.01, 1.12.01 XFERRED TO DIR. CALC, 3/9/93, 4/9/93 2320- PROFILES/factors OF INTEGERS 1-1,000,000, BY DS don.MCDONALD@nzpca.org.nz, don.lotto@paradise.net.nz WGTN, N.Z., (04) 389-6820 ENTER START NO. expression, 1 - N <= 2E9+, null = next, Q.UIT?25964900 later will come.. mersenne 42nd expon NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964900,6.74176032E14, 5095.58 2.2.5.5.13.19973 3.851353173 (1) 25623844 +341056 = 5062^2 + 584^2. (2) 25502500 +462400 = 5050^2 + 680^2. (3) 22052416 +3912484 = 4696^2 + 1978^2. (4) 19784704 +6180196 = 4448^2 + 2486^2. (5) 19360000 +6604900 = 4400^2 + 2570^2. (6) 13191424 +12773476 = 3632^2 + 3574^2. 25964901,6.74176084E14, 5095.58 3.3.3.961663 3.851353024 25964902,6.741761359E14, 5095.58 2.349.37199 3.851352876 25964903,6.741761878E14, 5095.58 151.373.461 3.851352727 25964904,6.741762397E14, 5095.58 2.2.2.3.7.7.22079 3.851352581 NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964950,6.741786286E14, 5095.58 2.5.5.11.17.2777 3.851345756 25964951,6.741786805E14, 5095.58 25964951 *** 3.851345608 42nd mersenne prime exponent. ******** 25964952,6.741787324E14, 5095.58 2.2.2.3.13.83221 3.851345459 25964953,6.741787843E14, 5095.58 7.7.23.23039 3.851345311 25964954,6.741788362E14, 5095.58 2.12982477 3.851345164 (1) 25959025 +5929 = 5095^2 + 77^2. NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964955,6.741788881E14, 5095.58 3.3.3.3.5.61.1051 3.851345016 25964956,6.7417894E14, 5095.58 2.2.59.269.409 3.851344867 25964957,6.741789919E14, 5095.58 25964957 3.851344719 (1) 20484676 +5480281 = 4526^2 + 2341^2. 25964958,6.741790438E14, 5095.58 2.3.4327493 3.85134457 25964959,6.74179096E14, 5095.58 25964959 3.851344422 NO. | SQUARE | SUM SQS | SQR | FACTORS | RECIP.| 25964960,6.741791479E14, 5095.58 2.2.2.2.2.5.7.97.239 3.851344272 PGM PROFILAC. E N D. *sp. ram.profono phon # off YES.Sun,27 Feb 2005.17:42:47 BBC Basic V Progm $..calc.factors.lgefactor.PrimeCabi, /6/94, /3/95 3.05.98 31.8.02, 5.01.04 BBC Basic V Progm $..calc.factors.lgefactor.PrimeCabi, /6/94, /3/95 3.05.98 31.8.02, 5.01.04 by Donald S McDonald, 63/3 Hutchison Rd, NEWTOWN, Wellington 2 New Zealand. Phone 64(NZ) +4(WN-) + 389-6820. Solve for (odd) factors, p <= 2^16.5 = 92,681. ABS( C * a^b -i ) <= n MOD p, Progm PrimeCabi multiplier * (base ^ index) + offset <= neighbourhood. Would you like to see some samples, examples? Yy ( N) ? PROC input. enter text, formula. Analysis of c, a, b, i, n later : ?m 25964951 multiplier * (base ^ index) + offset <= neighbourhood. 2E9 2E9 2E9 2E9 2^5. enter base ( expression a$ , max= 2E9, often 2 default, 10, 2^10, 1E3. ) ? a$ = = 2.000000000 enter index (power b%, [default 1]) ?25964951 enter multiplier, c% (default = 1) ? 1.000000000 enter offset, +/- i% = EVAL i$ (default 0) ? = 0.000000000 enter neighbourhood (sm. latitude) <= n%, (default = 0 ) ?35 find (first)factors p, satisfying , S% = m 25964951 *** ABS( 1.000000000 * 2.000000000 ^ 25964951.000000000 .. + 0.000000000 ) <= 35.000000000 MOD p. .... S% = 0 factor = 32768 = 2 * 16384 = 4 * 8192 = 8 * 4096 = 16 * 2048 = 32 * 1024 = 64 * 512 = 128 * 256 contin cs.= 1202 contin cs.= 1256 S% = -1 factor = 3 FIRST = 3 * 1PRIME. (test dup. factors). S% = -4 factor = 9 FIRST = 3 * 3 contin cs.= 1324 S% = 5 factor = 27 FIRST = 3 * 9 contin cs.= 1939 S% = 5 factor = 81 = 3 * 27 = 9 * 9 contin cs.= 2156 contin cs.= 2328 S% = -2 factor = 5 FIRST PRIME. (test dup. factors). S% = -2 factor = 25 = 5 * 5 contin cs.= 2391 S% = -2 factor = 125 = 5 * 25 contin cs.= 2473 contin cs.= 2537 S% = -3 factor = 7 FIRST PRIME. (test dup. factors). S% = -17 factor = 49 FIRST = 7 * 7 contin cs.= 2571 contin cs.= 2577 S% = -4 factor = 9 = 3 * 3 contin cs.= 2585 S% = 2 factor = 11 FIRST PRIME. (test dup. factors). S% = 2 factor = 121 = 11 * 11 contin cs.= 2595 contin cs.= 2601 S% = -6 factor = 13 FIRST PRIME. (test dup. factors). contin cs.= 2610 S% = -7 factor = 15 FIRST = 3 * 5 contin cs.= 2617 S% = -8 factor = 17 FIRST PRIME. (test dup. factors). contin cs.= 2627 .... S% = -11 factor = 4429 = 43 * 103 contin cs.= 4296 S% = -13 factor = 4493 PRIME. (test dup. factors). contin cs.= 4313 S% = 32 factor = 4519 PRIME. (test dup. factors). contin cs.= 4319 S% = 27 factor = 4729 PRIME. (test dup. factors). contin cs.= 4370 S% = -32 factor = 5021 PRIME. (test dup. factors). contin cs.= 4435 S% = -2 factor = 5101 PRIME. (test dup. factors). contin cs.= 4453 S% = -2 factor = 5237 PRIME. (test dup. factors). contin cs.= 4486 S% = 32 factor = 5419 PRIME. (test dup. factors). contin cs.= 4529 S% = 5 factor = 5443 PRIME. (test dup. factors). contin cs.= 4537 S% = 32 factor = 5461 = 43 * 127 contin cs.= 4541 S% = 4 factor = 5519 FIRST PRIME. (test dup. factors). contin cs.= 4555 S% = -2 factor = 5617 = 41 * 137 contin cs.= 4579 5733, S% = 2 factor = 5773 = 23 * 251 contin cs.= 4615 S% = 32 factor = 6403 = 19 * 337 contin cs.= 4762 S% = 31 factor = 6473 FIRST PRIME. (test dup. factors). contin cs.= 4779 S% = 32 factor = 6569 PRIME. (test dup. factors). contin cs.= 4802  Escape find (first)factors p, satisfying , S% = m 25964951 ABS( 1.000000000 * 2.000000000 ^ 25964951.000000000 + 0.000000000 ) <= 35.000000000 MOD p. highest factor tested, P% = 6569.000000000 cs. = 5223.000000000 at ERL. = 1430.000000000 summary factors, enter cont.? sumry factors, S% = 0 2 cont. sumry factors, S% = -1 3 cont. (MERSENNE ) sumry factors, S% = -2 5 S% = +2 11 cont. sumry factors, S% = -3 7 cont. sumry factors, S% = -4 9 S% = +4 5519 cont. sumry factors, S% = -5 0 S% = +5 27 cont. sumry factors, S% = -6 13 cont. sumry factors, S% = -7 15 S% = +7 67 cont. sumry factors, S% = -8 17 S% = +8 173 cont. , sumry factors, S% = -10 21 S% = +10 2609 cont. sumry factors, S% = -11 43 cont. sumry factors, S% = -12 0 S% = +12 47 cont. sumry factors, S% = -13 149 cont. sumry factors, S% = -14 1723 S% = +14 419 cont. sumry factors, S% = -15 0 S% = +15 277 cont. sumry factors, S% = -16 59 S% = +16 233 cont. sumry factors, S% = -17 49 S% = +17 271 cont. sumry factors, S% = -18 0 S% = +18 1171 cont. sumry factors, S% = -19 39 S% = +19 71 cont. sumry factors, S% = -20 0 S% = +20 677 cont. sumry factors, S% = -21 701 cont. sumry factors, S% = -22 45 cont. sumry factors, S% = -23 523 S% = +23 367 cont. sumry factors, S% = -24 139 cont. sumry factors, S% = -25 323 cont. sumry factors, S% = -26 53 S% = +26 493 cont. sumry factors, S% = -27 0 S% = +27 167 cont. , sumry factors, S% = -29 479 S% = +29 131 cont. , sumry factors, S% = -31 0 S% = +31 6473 cont. sumry factors, S% = -32 109 S% = +32 73 cont. sumry factors, S% = -33 107 cont. sumry factors, S% = -34 3499 cont. , BBC Basic V Program $.donmcd.calc.normal.PrimeCabi, e n d . 17/6/94, 5.01.04. $.bsiev8913 : enter base [retn 2], a = ? = 2 = Exponents start. reduced printout/ index expression (3.5E6/LOG 2) , return (NEXT = n lower case) 25964000 divisors of n-1 = 25964380 = 1*25964380/= 2*12982190/= 4*6491095/= 5*5192876/= 10*2596438/ = 13*1997260/= 20*1298219/= 26*998630/= 37*701740/= 52*499315/= 65*399452/ = 74*350870/= 130*199726/= 148*175435/= 185*140348/= 260*99863/ = 370*70174/= 481*53980/= 740*35087/= 962*26990/= 1924*13495/= 2405*10796/ = 2699*9620/= 4810*5398/-1 divisors of n-1 = 25964530 = 1*25964530/= 2*12982265/= 5*5192906/= 10*2596453/= 373*69610/ = 746*34805/= 1865*13922/= 3730*6961/-1 divisors of n-1 = 25964556 = 1*25964556/= 2*12982278/= 3*8654852/= 4*6491139/= 6*4327426/= 12*2163713/ = 521*49836/= 1042*24918/= 1563*16612/= 2084*12459/= 3126*8306/ = 4153*6252/-1 divisors of n-1 = 25964662 = 1*25964662/= 2*12982331/-1 divisors of n-1 = 25964842 = 1*25964842/= 2*12982421/= 71*365702/= 142*182851/-1 divisors of n-1 = 25964920 = 1*25964920/= 2*12982460/= 4*6491230/= 5*5192984/= 8*3245615/= 10*2596492/ = 20*1298246/= 40*649123/-1 divisors of n-1 = 25965160 = 1*25965160/= 2*12982580/= 4*6491290/= 5*5193032/= 8*3245645/= 10*2596516/ = 13*1997320/= 20*1298258/= 23*1128920/= 26*998660/= 40*649129/ = 46*564460/= 52*499330/= 65*399464/= 92*282230/= 104*249665/= 115*225784/ = 130*199732/= 167*155480/= 169*153640/= 184*141115/= 230*112892/ = 260*99866/= 299*86840/= 334*77740/= 338*76820/= 460*56446/= 520*49933/ = 598*43420/= 668*38870/= 676*38410/= 835*31096/= 845*30728/= 920*28223/ = 1196*21710/= 1336*19435/= 1352*19205/= 1495*17368/= 1670*15548/ = 1690*15364/= 2171*11960/= 2392*10855/= 2990*8684/= 3340*7774/ = 3380*7682/= 3841*6760/= 3887*6680/= 4342*5980/-1 divisors of n-1 = 25965256 = 1*25965256/= 2*12982628/= 4*6491314/= 8*3245657/= 17*1527368/ = 34*763684/= 68*381842/= 136*190921/-1 divisors of n-1 = 25965670 = 1*25965670/= 2*12982835/= 5*5193134/= 10*2596567/= 449*57830/ = 898*28915/= 2245*11566/= 4490*5783/-1 Escape17 at erl. 1290 PRIME n 25965769 ) close #0, 0 = YES/ contin 1 = NEXT ?0 enter neighbourhood (sm. latitude) <= n%, (default = 0 ) ?20 find factors p, satisfying , Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. bits h = 1 poss. area of improvement 1000^&F ? 2^h = 2 enter test divisors: start$ [Return = 1, .0 = quit] = ? start = 1 test unique even prime (plus powers 2) Spc = Yes, Nn no.? step 2*b% = 51929902 enter minimum [Return = 3] [N.B.** recommend BASIC 64.**] ? Sint = -1 factor*** = 155789707 PRIME. (test dup. factors). contin cs.= 60 find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 207719609 cs. = 971 at ERL. = 128 enter test divisors: start$ [Return = 1, .0 = quit] = ? start = 1 test unique even prime (plus powers 2) Spc = Yes, Nn no.? step 2 = 2 enter minimum [Return = 3] [N.B.** recommend BASIC 64.**] ? Sint = -1 factor = 3 = 3 * 1PRIME. (test dup. factors). , Sint = -4 factor = 9 = 3 * 3 contin cs.= 13 Sint = 5 factor = 27 = 3 * 9 contin cs.= 795 Sint = 5 factor = 81 = 3 * 27 = 9 * 9 contin cs.= 1162 , contin cs.= 1242 Sint = -2 factor = 5 PRIME. (test dup. factors). , Sint = -2 factor = 25 = 5 * 5 contin cs.= 1330 Sint = -2 factor = 125 = 5 * 25 contin cs.= 1474 , contin cs.= 1774 Sint = -3 factor = 7 PRIME. (test dup. factors). , Sint = -17 factor = 49 = 7 * 7 contin cs.= 1859 , contin cs.= 1926 Sint = -4 factor = 9 = 3 * 3 contin cs.= 2005 Sint = 2 factor = 11 PRIME. (test dup. factors). , Sint = 2 factor = 121 = 11 * 11 contin cs.= 2088 , contin cs.= 2165 Sint = -6 factor = 13 PRIME. (test dup. factors). , contin cs.= 2253 Sint = -7 factor = 15 = 3 * 5 contin cs.= 2324 Sint = -8 factor = 17 PRIME. (test dup. factors). , contin cs.= 2407 Sint = -6 factor = 19 PRIME. (test dup. factors). , Sint = -6 factor = 361 = 19 * 19 contin cs.= 2501 , contin cs.= 2526 Sint = -10 factor = 21 = 3 * 7 contin cs.= 2534 Sint = 2 factor = 23 PRIME. (test dup. factors). , contin cs.= 2549 Sint = -2 factor = 25 = 5 * 5 contin cs.= 2556 Sint = 5 factor = 27 = 3 * 9 contin cs.= 2563 Sint = -3 factor = 29 PRIME. (test dup. factors). , contin cs.= 2576 Sint = 2 factor = 31 PRIME. (test dup. factors). , contin cs.= 2589 Sint = 2 factor = 33 = 3 * 11 contin cs.= 2596 Sint = -17 factor = 35 = 5 * 7 contin cs.= 2603 Sint = 5 factor = 37 PRIME. (test dup. factors). , contin cs.= 2616 Sint = -19 factor = 39 = 3 * 13 contin cs.= 2623 Sint = -2 factor = 41 PRIME. (test dup. factors). , contin cs.= 2637 Sint = -11 factor = 43 PRIME. (test dup. factors). , contin cs.= 2650 Sint = 12 factor = 47 PRIME. (test dup. factors). , contin cs.= 2669 Sint = -17 factor = 49 = 7 * 7 contin cs.= 2676 Sint = -16 factor = 59 PRIME. (test dup. factors). , contin cs.= 2696 Sint = 7 factor = 67 PRIME. (test dup. factors). , contin cs.= 2716 Sint = 19 factor = 71 PRIME. (test dup. factors). , contin cs.= 2729 Sint = 2 factor = 89 PRIME. (test dup. factors). , contin cs.= 2761 Sint = -2 factor = 101 PRIME. (test dup. factors). , contin cs.= 2781 Sint = -11 factor = 103 PRIME. (test dup. factors). , contin cs.= 2794 Sint = -2 factor = 137 PRIME. (test dup. factors). , contin cs.= 2839 Sint = -13 factor = 149 PRIME. (test dup. factors). , contin cs.= 2859 Sint = 8 factor = 173 PRIME. (test dup. factors). , contin cs.= 2897 Sint = -4 factor = 179 PRIME. (test dup. factors). , contin cs.= 2911 Sint = 16 factor = 233 PRIME. (test dup. factors). contin cs.= 2974 Sint = 2 factor = 251 PRIME. (test dup. factors). contin cs.= 2994 Sint = 17 factor = 271 PRIME. (test dup. factors). contin cs.= 3020 Sint = 15 factor = 277 PRIME. (test dup. factors). contin cs.= 3027 Sint = -6 factor = 361 = 19 * 19 contin cs.= 3127 Sint = -2 factor = 397 PRIME. (test dup. factors). contin cs.= 3172 Sint = 14 factor = 419 PRIME. (test dup. factors). contin cs.= 3192 Sint = 8 factor = 431 PRIME. (test dup. factors). contin cs.= 3205 Sint = 2 factor = 601 PRIME. (test dup. factors). contin cs.= 3410 Sint = 20 factor = 677 PRIME. (test dup. factors). contin cs.= 3504 Sint = 2 factor = 683 PRIME. (test dup. factors). contin cs.= 3511 Sint = 2 factor = 713 = 23 * 31 contin cs.= 3549 Sint = -13 factor = 877 PRIME. (test dup. factors). contin cs.= 3736 Sint = 2 factor = 881 PRIME. (test dup. factors). contin cs.= 3743 Sint = -2 factor = 953 PRIME. (test dup. factors). contin cs.= 3831 Sint = -2 factor = 1021 PRIME. (test dup. factors). contin cs.= 3919 Sint = 16 factor = 1103 PRIME. (test dup. factors). contin cs.= 4019 Sint = 18 factor = 1171 PRIME. (test dup. factors). contin cs.= 4095 1513 cs.= 4496,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Sint = -14 factor = 1723 PRIME. (test dup. factors). contin cs.= 4761 Sint = 2 factor = 1801 PRIME. (test dup. factors). contin cs.= 4849 Sint = 2 factor = 1871 PRIME. (test dup. factors). contin cs.= 4931 Sint = 2 factor = 2047 = 23 * 89 contin cs.= 5141 Sint = 16 factor = 2089 PRIME. (test dup. factors). contin cs.= 5205 Sint = -2 factor = 2113 PRIME. (test dup. factors). contin cs.= 5224 2459 cs.= 5626,,,,,,,,,,,,,,,,,,, Sint = -6 factor = 2557 PRIME. (test dup. factors). contin cs.= 5750 Sint = 10 factor = 2609 PRIME. (test dup. factors). contin cs.= 5807 Sint = -14 factor = 2741 PRIME. (test dup. factors). contin cs.= 5969 Sint = 2 factor = 2759 = 31 * 89 contin cs.= 5995 Sint = 2 factor = 2971 PRIME. (test dup. factors). contin cs.= 6249 Sint = 2 factor = 3191 PRIME. (test dup. factors). contin cs.= 6503 3539 cs.= 6905,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Sint = -4 factor = 3739 PRIME. (test dup. factors). contin cs.= 7158 Sint = 2 factor = 4051 PRIME. (test dup. factors). contin cs.= 7530 Sint = -2 factor = 4141 = 41 * 101 contin cs.= 7636 Sint = 15 factor = 4217 PRIME. (test dup. factors). contin cs.= 7718 Sint = -4 factor = 4273 PRIME. (test dup. factors). contin cs.= 7805 Sint = -2 factor = 4421 PRIME. (test dup. factors). contin cs.= 7967 Sint = -11 factor = 4429 = 43 * 103 contin cs.= 7987 Sint = -13 factor = 4493 PRIME. (test dup. factors). contin cs.= 8069 4843 cs.= 8471,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Sint = -2 factor = 5101 PRIME. (test dup. factors). contin cs.= 8773 Sint = -2 factor = 5237 PRIME. (test dup. factors). contin cs.= 8941 Sint = 5 factor = 5443 PRIME. (test dup. factors). contin cs.= 9190 Sint = 4 factor = 5519 PRIME. (test dup. factors). contin cs.= 9278 Sint = -2 factor = 5617 = 41 * 137 contin cs.= 9396 Sint = 2 factor = 5773 = 23 * 251 contin cs.= 9577 6109 cs.= 9979,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 6373 cs.= 10300,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 6649 cs.= 10614,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 6911 cs.= 10929,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 7177 cs.= 11243,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 7453 cs.= 11558,,,,,, Sint = 2 factor = 7481 PRIME. (test dup. factors). contin cs.= 11602 Sint = 2 factor = 7781 = 31 * 251 contin cs.= 11961 Sint = -7 factor = 8087 PRIME. (test dup. factors). contin cs.= 12320 Sint = -2 factor = 8101 PRIME. (test dup. factors). contin cs.= 12340 8441 cs.= 12742,,,,,,,,,,, Sint = -2 factor = 8501 PRIME. (test dup. factors). contin cs.= 12817 8843 cs.= 13218,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9109 cs.= 13532,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9377 cs.= 13847,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9641 cs.= 14161,,,,,,,,,,,,,, Sint = 8 factor = 9719 PRIME. (test dup. factors). contin cs.= 14254 find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 9953 cs. = 14542 at ERL. = 129 ... cs.= 13847,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 9641 cs.= 14161,,,,,,,,,,,,,, Sint = 8 factor = 9719 PRIME. (test dup. factors). contin cs.= 14254 find factors p, satisfying Sint = M-25964951 ABS( 1 * 2 ^ 25964951 + 0 ) <= 20 MOD p. highest factor tested, P% = 9953 cs. = 14542 at ERL. = 129 ... enter test divisors: start$ [Return = 1, .0 = quit] = ?0 ENTER START NO. expression, 1 - N <= 2E9+, null = next, (1) Solve for (odd) factors, p <= 2^16.5 = 92,681. ABS( C * a^b -i ) <= n MOD p, Progm PrimeCabi multiplier * (base ^ index) + offset <= neighbourhood. multiplier * (base ^ index) + offset <= neighbourhood. enter base ( expression a$ , max= 2E9, often 2 default, 10, 2^10, 1E3. ) ? a$ = = 2.000000000 enter multiplier, c% (default = 1) ? enter offset, +/- i% = EVAL i$ (default 0) ? = 0.000000000 enter neighbourhood (sm. latitude) <= n%, (default = 0 ) ?35 find (first)factors p, satisfying , S% = m 25964951 ABS( 1.000000000 * 2.000000000 ^ 25964951.000000000 + 0.000000000 ) <= 35.000000000 MOD p. S% = 0 factor = 16384 = 2 * 8192 = 4 * 4096 = 8 * 2048 = 16 * 1024 = 32 * 512 = 64 * 256 = 128 * 128 contin cs.= 1149 S% = 0 factor = 32768 = 2 * 16384 = 4 * 8192 = 8 * 4096 = 16 * 2048 = 32 * 1024 = 64 * 512 = 128 * 256 contin cs.= 1202 contin cs.= 1256 S% = -1 factor = 3 FIRST = 3 * 1PRIME. (test dup. factors). S% = -4 factor = 9 FIRST = 3 * 3 contin cs.= 1324 S% = 5 factor = 27 FIRST = 3 * 9 contin cs.= 1939 S% = 5 factor = 81 = 3 * 27 = 9 * 9 contin cs.= 2156 contin cs.= 2328 S% = -2 factor = 5 FIRST PRIME. (test dup. factors). S% = -2 factor = 25 = 5 * 5 contin cs.= 2391 S% = -2 factor = 125 = 5 * 25 contin cs.= 2473 contin cs.= 2537 S% = -3 factor = 7 FIRST PRIME. (test dup. factors). S% = -17 factor = 49 FIRST = 7 * 7 contin cs.= 2571 contin cs.= 2577 S% = -4 factor = 9 = 3 * 3 contin cs.= 2585 S% = 2 factor = 11 FIRST PRIME. (test dup. factors). S% = 2 factor = 121 = 11 * 11 contin cs.= 2595 contin cs.= 2601 S% = -6 factor = 13 FIRST PRIME. (test dup. factors). contin cs.= 2610 S% = -7 factor = 15 FIRST = 3 * 5 contin cs.= 2617 S% = -8 factor = 17 FIRST PRIME. (test dup. factors). contin cs.= 2627 ... S% = 5 factor = 5443 PRIME. (test dup. factors). contin cs.= 4537 S% = 32 factor = 5461 = 43 * 127 contin cs.= 4541 S% = 4 factor = 5519 FIRST PRIME. (test dup. factors). contin cs.= 4555 S% = -2 factor = 5617 = 41 * 137 contin cs.= 4579 S% = 2 factor = 5773 = 23 * 251 contin cs.= 4615 S% = 32 factor = 6403 = 19 * 337 contin cs.= 4762 S% = 31 factor = 6473 FIRST PRIME. (test dup. factors). contin cs.= 4779 S% = 32 factor = 6569 PRIME. (test dup. factors). contin cs.= 4802 ... X-RFC2646: Format=Flowed; Original don.mcdonald 27.02.05 23:23 23:49 SENT. Yeah, what he said. Jay X-RFC2646: Format=Flowed; Response The answer is 11, isn't it?? Hello All, Take the following equation: y^2 = (1+x^2)/2 If solved for whole numbers, the first obvious pair is x_1=1, y_1=1. The second is not hard to find: x_2=7, y_2=5 If you go on, you end up with something like this: using k=6 (which seems to be equal to k = x_(n) -x_(n-1) = y_(n)+y_(n+1)) y_(n) = y_(n-1)*k + y_(n-2) (for n=3,...) x_(n) = x_(n-1)*k - x_(n-2) (for n=3,...) using the equation: x_3 = 6*x_2 - x_1 = 6*7-1 = 41 y_3 = 6*y_2 - y_1 = 6*5-1 = 29 x_4 = 6*x_3 - x_2 = 6*41-7 = 239 y_4 = 6*y_3 - y_2 = 6*29-5 = 169 so on. Is this something known? The y_n's are recognisable as the hypotenuses of successive almost isosceles right triangles - their shorter sides differ by 1 and sum to the corresponding x_n: y_1: 0, 1, 1 y_2: 3, 4, 5 y_3: 20, 21, 29 y_4: 119, 120, 169 -- Paul Townsend Pair them off into threes Interchange the alphabetic letter groups to reply This means that x/y must approximate sqrt(2) quite closely. Good approximations for sqrt(2) can be obtained by truncating its 1/1 3/2 7/5 17/12 41/29 99/70 239/169 577/408 1393/985 ... Maybe you'll recognize some of these? The other ones satisfy y^2 = (-1+x^2)/2 which is quite similar to your equation. Recurrence relations can be derived for these regular continued fractions, including yours. See e.g. http://www.daviddarling.info/encyclopedia/S/square_root_of_2.html So, not new, but nicely found! -- M.vr.gr. Dave Langers (Voor mail, vervang voornaam en achternaam) yazdi:cvvkqr$d41$1@domitilla.aioe.org... OK. Actually it is special form of the following. y^2 = (n-1+x^2)/n So for eample n=5 y^2 = (4+x^2)/5 x_1 = 1, y_1=1 x_2 = 4, y_2=2 k=3 (=1+2=4-1) x_n = k*x_(n-1)-x_(n-2) (for n=3,...) y_n = k*y_(n-1)-y_(n-2) (for n=3,...) x_3 = 3*x_2 - x_1 = 3*4-1= 11 y_3 = 3*y_2 - y_1 = 3*2-1 = 5 x_4 = 3*x_3-x_2=3*11-4=29 y_4 = 3*y_3-y_2=3*5-2=13 x_5 = 3*29-11=76 y_5 = 3*13-5=34 so on. Similarly they can be used as an approximation for sqrt(5) then: 1/1 4/2 11/5 29/13 76/34 Search with keyword: Pell's equation You will get about 5,000 hits. Hit #5 is: http://mathworld.wolfram.com/PellEquation.html Hit #2 is: http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pell.html yazd.88:luudneAXjpQ8zr7fRVn-2Q@comcast.com... OK. I didn't know that. No. You're the first person on the planet to have ever noticed that. I suggest you start applying for your Nobel Award pronto! X-RFC2646: Format=Flowed; Response ----- Original Message ----- {skip}] &&& Dave are you trying to be nice and helpful? If so, please explain the Nobel Award for Mathematics. I never heard of that. Could a person really apply for it? - Dan in NY (for email change t with g in dKlinkenbert at hvc dot rr dot com) X-RFC2646: Format=Flowed; Original I would be very grateful if someone could help me with this. x=0 y-z+1=0 other way to write this is: *p1....x/0=y/1=(z-1)/1 now we searh the thing which is writen in this formula: Ax+By+Cz+D=0...this is flat area pi... this flat area is _I_ (90degrees) to x0y flat area. and this flat area has in it another linear form..like p1*. it has p2. this p2 is _I_ (90deg) to p1. spot (1,-1, 1) belongs to p1, and also to flat area pi. thank you very much.