20090504, 16:16  #1 
May 2005
74_{8} Posts 
Odd perfect related road blocks II
Alex, fivemack, ATH, Andi47, Andi HB & all other factoring specialists,
Let M be a smallest product of odd primes such that square free of ( gcd ( f(x) ,g(x)) )  M for all integer x > 1 , where f(x) and g(x) are two given relatively prime integral polynomials. For example, if f(x) = (x^{2} + x + 1)^{3}  3^{3} & g(x) = x^{6} + x^{3} + 1 = 0, then M = 3 * 19 * 163 if f(x) = ((x^{4} + x^{3} + x^{2} + x + 1)^{5}  5^{5} ) /(x^{4} + x^{3} + x^{2} + x + 1  5 ) & g(x) = x^{20} + x^{15} + x^{10} + x^{5} + 1, then M = 5 * 151 * 701 * 2551 * 24251 * 34651 * 144853351 * 659575601 * 1271785993801. By applying successive (my) extended Eculid algorithms, I have encountered if 1) f(x) = (x^{2} + x + 1)^{3} – 3^{3} & g(x) = (x^{162} + x^{81} + 1)^{3} – 3^{3} , then Code:
M = a * 598644007623271861103907634394401713031280184082045961436816127153183697504366019739512180572430920157281903457051081235141165004401529332661832140639096711933077 2) f(x) = ((x^{2} + x + 1)^{9}  3^{9} ) / ((x^{2} + x + 1)^{3}  3^{3} ) & g(x) = x^{162} + x^{81} + 1 then Code:
M = b * 142255953422080949010042135770532701808988159571292214113059412169543088033374620681986196789531788565741609436257609136785819948534868895759387087438788760412860551 where a and b are products of known primes, c162 and c165 are two critical roadblocks I am unable to complete by myself. Note that: these M‘s are part of my “pushing “ algorithms concerning several ongoing papers (first one is “On the divisibility of systems of cyclotomic polynomials of degrees 3 and 5”) . Alex may be familiar with c162 and c165 at certain degrees. I sincerely hope it will get completed this time, otherwise one of my lemmas will get killed. Can someone please kindly look into it or give a try before I finally give up. I also found c1.x + c0 = 0 (mod c162), where Code:
c1 = 69267298939868734352823569242638987830151794956582757323663257304713979808535979493103174703607420534575191258096981359057081634837910521017237222985183216764829376 c0 = 90396638571991941300717867224156466586094273057103180468944748956569143785197229469169728666741827825163688170928805513389627480981342184801410418423802620496945754 Code:
c1 = 76753267437136028631737826218354419681592252499734800494051831532232913505065057592910545783967666343514329739056408237901354351164460723435804959401745218854773 c0 = 209851117037629231974563829915727038069629516190909815572098319849496046563710374917531353670135870468192288676994164729691076296875290909453103109496164911219198 I am looking for one factor(only) from each of the following. Some of them may have posted before *, although these composites did not affect the proof, they caused me a lot of unnecessary work, please help and thank you in advance. 1) σ(59345478426821800746377014559617^4) = 3538361.c121 # 2) σ(99995282631947^10) = 23.c139 3) σ(62060021^18) = c141 * =2859153813495302135105360393 * 65219432427202213218611042380245134951516556220905814475152737256300473009279433432471898046319778085284772450023 (mklasson, ecm) 4) σ(347568611538691^10) = 23.617.683.692539.c133 5) σ(10177286401^12) =15731.c116 * 6) σ(6294091^22) = c150 * 7) σ(5229043^22) = 967.194443.c140 * 8) σ(846041103974872866961^6) =1709.c123 = 1709 . 5155723754893919994283900206097487201 . 41621437657162669816374890034224257842979191889517024692601717156897218627524115006963 (fivemack, ecm) 9) σ(934415109937^12) = 2549.c141 = 2549 . 2636398433195487889353395293207 . 65932167085234904931409415588486535060018564833464183835891750325639105962353333551790922471214875173575350927 (mklasson, ecm) 10) σ(P82^2) = 61.433.4651.156817.c150 *, where P82= 3404253904642598840161913302701587626837449819596812423232352 24442734524 2057474631 11) σ(797161^28) = 59.31727.c159 12) σ(172545754771028210096747645881^6) = 631.c173 13) σ(172827552198815888791^6) = 113.c120 = 113 . 3335162523320119257180081115662029635417481 . 70710259042622884083135682057512365547361929153437894550723481191426825243649 (fivemack, snfs) 14) σ(177635683940025046467781066894531^4) = 5.431.c126 *  most wanted &  Alex resolved half a dozen and wblipp got one last year. #  I completed at least several hundreds or even thousand curves per each composite a year ago but lost the record. Best regards Joseph Last fiddled with by fivemack on 20090505 at 00:30 
20090504, 17:02  #2 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Joseph,
I made some typographical changes to fix the exponents and to avoid horizontal scrolling. Can you please check that I didn't introduce any errors? Alex 
20090504, 17:08  #3 
Nov 2008
2·3^{3}·43 Posts 
You seem to have missed a p11: 68280416627. It leaves a c110.
These are both difficulty ~120 via a sextic. I suggest you factor them yourself. They should be easy. Last fiddled with by 10metreh on 20090504 at 17:14 
20090504, 17:22  #4 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 

20090504, 17:25  #5 
Feb 2004
2×3×43 Posts 

20090504, 17:26  #6  
Nov 2008
2×3^{3}×43 Posts 
Quote:
(Yes, you are more experienced than me, so you are giving me advice if you correct me.) And about the p11 issue: I'm not sure. I probably had a typo in the input. Edit: found the typo. I missed out a digit in the number Last fiddled with by 10metreh on 20090504 at 17:34 

20090504, 17:41  #7 
Feb 2004
2×3×43 Posts 
p31=2636398433195487889353395293207  sigma(934415109937^12) / 2549 with cofactor p110.

20090504, 17:48  #8 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Ok, I understand that you want to factor the resultant of f(x) and g(x), for example
Code:
? f1(x) = (x^2+x+1)^3  3^3 ? g1(x) = x^6 + x^3 + 1 ? factorint(polresultant(f1(x), g1(x))) %15 = [3 6] [19 3] [163 1] ? f2(x) = (((x^51)/(x1))^55^5) / (x^4 + x^3 + x^2 + x + 1  5) ? g2(x) = (x^251)/(x^51) ? factorint(polresultant(f2(x), g2(x))) %16 = [5 16] [151 1] [701 1] [2551 1] [24251 1] [34651 1] [144853351 1] [659575601 1] [1271785993801 1] Code:
? f3(x) = (x^2+x+1)^33^3 ? g3(x) = (x^162+x^81+1)^33^3 ? gcd(f3(x), g3(x)) %23 = x  1 ? polresultant(f3(x)/(x1),g3(x)/(x1)) <big highly composite number, has your c162 as divisor> Code:
? f4(x)=((x^2+x+1)^93^9)/((x^2+x+1)^33^3) ? g4(x)=x^162 + x^81 + 1 ? polresultant(f4(x),g4(x)) <big highly composite number, has your c165 as divisor> Alex 
20090504, 17:52  #9 
Feb 2004
2·3·43 Posts 
sigma(62060021^18):
********** Factor found in step 2: 2859153813495302135105360393 Found probable prime factor of 28 digits: 2859153813495302135105360393 Probable prime cofactor 65219432427202213218611042380245134951516556220905814475152737256300473009279433432471898046319778085284772450023 has 113 digits 
20090504, 17:52  #10  
"Nancy"
Aug 2002
Alexandria
9A3_{16} Posts 
Quote:
I thought your telling Joseph off ("I suggest you factor them yourself.") wasn't called for. Alex 

20090504, 18:04  #11  
Nov 2008
2322_{10} Posts 
Quote:
Anyway: I managed to get the polynomials x^6 + x^5  5x^4  4x^3 + 6x^2 + 3x  1 and 10177286401x + 103577158487979532802 using phi (yes, I know that is your program). I presume the method used is similar to the one mentioned on the wiki for x^13k1. Last fiddled with by 10metreh on 20090504 at 18:15 

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