[QUOTE=3.14159;236590]Here's a nice challenge for you all;

Express the number 988741313296003[sup]47[/sup] as a sum of prime powers, which meet these conditions:

1. No more than 25 primes may be used.

Ex: 2[sup]30[/sup] + 3[sup]18[/sup] + 5[sup]12[/sup] + 19[sup]6[/sup] + 293[sup]3[/sup] + 577[sup]3[/sup] + 191[sup]3[/sup] + 89[sup]3[/sup] + 223[sup]2[/sup] + 37[sup]2[/sup] + 11[sup]2[/sup] + 73[sup]1[/sup] + 2[sup]1[/sup] = 7[sup]11[/sup]

Or, for a harder one; No primes smaller than 250000 may be used, no primes may be used more than once, all exponents must be prime and used once and only once, and no more than 30 primes may be used.[/QUOTE]

if only I knew how to implement the four squares theorem, or the theorem that states every number can be expressed as a sum of 19 fourth powers in such a way to use it to figure out a way to represent the base with prime powers I could maybe do it lol.

pi your base is prime hence no configuring needs to be done for 1. as for 2 I'll try it out maybe lol. oh wait if 1 prime base and exponent pair is needed you gave it stating the question my work is done. okay maybe not as the 250000 part you're saved from completely being destroyed by me.

[QUOTE=3.14159;236590]Express the number 988741313296003[sup]47[/sup] as a sum of prime powers, which meet these conditions:

1. No more than 25 primes may be used.

Ex: 2[sup]30[/sup] + 3[sup]18[/sup] + 5[sup]12[/sup] + 19[sup]6[/sup] + 293[sup]3[/sup] + 577[sup]3[/sup] + 191[sup]3[/sup] + 89[sup]3[/sup] + 223[sup]2[/sup] + 37[sup]2[/sup] + 11[sup]2[/sup] + 73[sup]1[/sup] + 2[sup]1[/sup] = 7[sup]11[/sup]

Or, for a harder one; No primes smaller than 250000 may be used, no primes may be used more than once, all exponents must be prime and used once and only once, and no more than 30 primes may be used.[/QUOTE]

Well, 988741313296003[sup]47[/sup] is a solution to both, so I imagine you intended to disallow that.

For the first, the simplest remaining solution is x[sup]1[/sup] + 5441[sup]1[/sup] + 3[sup]1[/sup].

The second seems hard. I assume that the size limit on primes does not apply to exponents, or else the problem has no solutions. It's not clear whether you're limiting the number of summands to 15 or 30.

[QUOTE=science_man_88;236597]pi your base is prime hence no configuring needs to be done for 1.[/QUOTE]

You beat me to it!

I think I might have been able to phrase that a bit better..

[QUOTE=CRGreathouse;236604]You beat me to it![/QUOTE]

I was in the thread just after he posted it and and had pari open so I tried it out lol.

Okay; Try answering this;

Can every integer > 10[sup]6[/sup] be expressed as a sum of prime squares?

Example: 1802042 = 1009^2 + 877^2 + 113^2 + 43^2 + 13^2 + (3^2) * 5

Another; 545014337 = 23117^2 + 3167^2 + 761^2 + 97^2+ 11^2 + (2^2)*18 + (3^2)*4

[QUOTE=3.14159;236735]Can every integer > 10[sup]6[/sup] be expressed as a sum of prime squares?

Example: 1802042 = 1009^2 + 877^2 + 113^2 + 43^2 + 13^2 + (3^2) * 5

Another; 545014337 = 23117^2 + 3167^2 + 761^2 + 97^2+ 11^2 + (2^2)*18 + (3^2)*4[/QUOTE]

Yes. In fact, this holds for every integer greater than 23. (To prove this it suffices to prove it for {24, 25, 26, 27}. Alternately, a method of Sylvester (1884) allows a one-step proof for n > 35.)

If you meant "distinct prime squares", then the answer is still yes, though I can't give an elementary proof. (A proof is not difficult with a little bit of computer power plus a weak bound on the growth of primes, e.g., p[sub]n+1[/sub] < 1.4p[sub]n[/sub].) The largest number that can't be so expressed is 17,163.

Part 2 of; Factoring random 100-digit numbers;

1000309121763716018192725963203643789700149527936906867199570524663892824104343656283438202733442867 = 3 * 827 * 72245231489 * 15956241661 * p32 * p43

5368323813249515329098296538291057500943044787302529648340980984764463865258985384399553193181390034 = 2 * 3[sup]2[/sup] * 17 * 29 * 113 * 1218 * 51503 * 141397 * p81

8013176147722647871956980350400065506099323485506514645505443741520820451558115945129175079407373895 = 5 * p22* p33* p46

1152443181169586383493536135294025497382757172618548038480830431161568631322294266650168720443874455 = 5 * p99

7666551255558602189678572939665486145558049206656581150748267177043910125431974118219650818960910581 = 59 * p99

2562808938752647864331131924722585579690624807298066926175303729176944775263748281662129402089103030 = 2 * 5 * 22153 * 19445759572794150941 * p75

8199342366679060827617296146239673154845942561339379994399286622024273611547154072068685152481812583 = 7 * 163 * 69149 * 133831 * p87

3817078147299184807266522855781210143244534280399726146655814305698076033398396433478794563004496650 = 2 * 5[sup]2[/sup] * 7193 * p21 * p74

2483514470656621527509543660684300711436807489222617550372160227312222630960322334656545109061468245 = 5 * 7 * 53 * p20 * p77.

5342453719772700595767968882054794877986643318570295675203507607631997227930905300848205513566413849 = 129749 * p95.

There you have it, ten 100-digit numbers, completely factored.

Pfft, here's 100.

Oops, the forum won't let me post that many characters. OK, here's the amount I can fit into a post.

[code]3828341080244400637832496477959805704164073468183074462634180957833651864984723682138669035433026500 = 2 * 2 * 3 * 5 * 5 * 5 * 7 * 19 * 19 * 23 * 37 * 53 * 241 * 1181 * 5649913 * 36081377 * 8581181173 * 47840420649221 * 56402348128571 * p32
1588660322983396302885796245830814329302488843364165698215578492464397730424401789229489704601075756 = 2 * 2 * 3 * 7 * 13 * 23 * 47 * 61 * 181 * 15031 * 526483 * 927323 * 1148527 * 406979061818023 * 81538217474440208479553 * p30
1021031048497837374483531447808891337935989532331588576367878602517111431033593224173329292881512173 = 31 * 367 * 30809 * 5668727 * 908392703 * 1028314699 * 2197843586849586383840467 * p42
7086422853097868451711345316509410180470376785600011857778722004533443743419826800885302936650498470 = 2 * 5 * 13 * 19 * 29 * 431 * 587 * 691 * 2767 * 942187 * 1893539 * 2597779 * 15624896897 * 1610494874908351 * 3845935534903586029 * p21
4318456126006444342857842557107114096633481415982568386769884274414806491142401044261345969736720694 = 2 * 3 * 7 * 11 * 13 * 23 * 31159 * 302053 * 30936197387 * 4176718791229 * 847849456387121 * 2254092997188947242409 * p26
1109587849660304353836201770893761411871650688020685001076825645933095103648972795052962839552595690 = 2 * 3 * 3 * 5 * 7 * 23 * 31 * 67 * 163 * 85632998041 * 17249535181483504313 * 394021370780626750477817951 * p33
2979941477970790621634249796909439789746784691205449811130412558873271515382744651517007594645175894 = 2 * 3 * 11 * 13 * 19 * 97 * 1637 * 706109 * 60265063 * 59542983281711851 * 6457175558840031600628637783 * p32
6420726987540740841706278258860741277295366130264261913963063511707914602848605467111390751821088121 = 3 * 7 * 97 * 1481121042485345479125151 * 176615229066307992468364472796139 * p41
9816138939953536743143344071912358383004405759221258712071041553368290365584815105024598168207976103 = 839 * 10399 * 4264780100791321147 * 8561427410274222444303852083 * p47
2268168812216914591854154823153328155076179646049376828112834528320073183563365268538263313054628020 = 2 * 2 * 3 * 3 * 5 * 7 * 7 * 23 * 31 * 31 * 37 * 61 * 73 * 479 * 523 * 1486553277901 * 2312246873086067716693912462289027 * p35
2175187002194884841675973456689521692815926892066030053856510088603896308976058660302016139102867286 = 2 * 17 * 43 * 199 * 317 * 17207 * 595611953 * 374487601672675901611 * 120397871923043244929719 * p35
1964666465908214969730131549122817558003317961763724416749317778247127635667033822930173507293655037 = 1409 * 2621779 * 572550962994499639130106764801815763499954743 * p45
2109835988366518432588338355806050669474506758795640108970282822027494388216935165502614463059840500 = 2 * 2 * 3 * 3 * 5 * 5 * 5 * 13 * 17 * 53 * 53 * 89 * 241 * 279464069 * 13390332799 * 13566564771765494922641095631 * p39
4170436258982883679101363076894140796229193134975892831177906516068402234097418053129515314672448236 = 2 * 2 * 17 * 2069 * 131381 * 147541 * 392669 * 1385569 * 8654839 * 1817213413 * 641105533842841 * 22777900103786411 * p26
1112745709173761896559835695316280609014296134651327666778801370828688600198148161143564180857076834 = 2 * 3 * 7 * 11 * 31 * 47 * 317 * 19381 * 182617 * 304849 * 589163046401 * 490279242759107592236447059 * p38
6754467728048555661382638518307179841909700849996023430663298769830785246937785624396484840143801820 = 2 * 2 * 5 * 13 * 19 * 23 * 29 * 83 * 97 * 2029 * 102139 * 175919 * 28920977 * 37222903 * 3325507556831 * 113026544095967 * p35
9290303292654677796208450313997479786862212010805900056412444714830815535310109739094853648011955880 = 2 * 2 * 2 * 3 * 5 * 19 * 19 * 53 * 139 * 149 * 2633 * 254557 * 7518315653802366207934294760412271 * p47
1519045907326352179272495192012691255110343871460740344723603485053123113376600850827082587042722632 = 2 * 2 * 2 * 3 * 3 * 3 * 7 * 7 * 7 * 7 * 11 * 13 * 19 * 23 * 73 * 953 * 8039 * 191531 * 650543 * 966923 * 2910837913 * 4712082339471169543 * p35
2297494279297747651318689357388091486187167887028748530651383542166405596399953892245633133215771067 = 3 * 19 * 1129 * 4909 * 45281 * 95483 * 350573551 * 185546423887211 * 5000505715976357 * p43
6916716710106028694497858360968629919200917839743569117166581989808297983518264029006195056926115135 = 3 * 5 * 7 * 313 * 599 * 701 * 887 * 117331 * 4657853 * 8510059 * 3519961031 * 34327729151 * 346541139438472730910749 * p25
4020646510893532589104720126916970485917983107805030705775412066914614392060823406100284080206300400 = 2 * 2 * 2 * 2 * 3 * 3 * 5 * 5 * 7 * 7 * 7 * 13 * 31 * 31 * 43 * 67 * 349 * 33623 * 102679 * 142159 * 229717 * 10307723 * 338788421 * 1427600611 * p39
7172656039606291939374148166929125456455438300989478240428912337825056979098115354643207581855835285 = 3 * 3 * 5 * 7 * 13 * 19 * 37 * 61 * 487 * 1121941560287 * 121409626878654282391474088573 * p48
2528919604826729658531053105058688132708635900874626524727189778747923659829992520183389956689288493 = 7 * 19 * 37 * 127 * 887 * 2347 * 149551999 * 140397312799 * 46097360176928611953517733 * p43
4780580308922684791934717889134818971136050278964890263719923034086360164542392367179511433499581230 = 2 * 3 * 5 * 11 * 47 * 107 * 199 * 3559 * 101399 * 226307 * 1130190559 * 12732542650017211591 * 130385357953530112759 * p29
1771581299718052268780183255753188184264364543991851960093639623289282075054030826702670725925580126 = 2 * 3 * 7 * 23 * 73 * 1621 * 440681 * 57252901 * 9424402487 * 555590203710791497 * 1664351435128005575192027 * p26
1826607301881605308867617623466131387321007525492908119055581832481636954149723135560049573051779689 = 3 * 7 * 8731 * 11971 * 86423 * 133697 * 6175693 * 3399940349409841 * 228101887805109423382486397 * p32
5012088009277640314402761559890041839759396057018545897258241994190447331558998944288170990344028751 = 2917 * 1757549 * 85744849549 * 77390355798296212981 * 117857840533006506831097561 * p34
1118682071008816168222725102230550120702126996309533995481998016819660387118847018742147190609638748 = 2 * 2 * 3 * 7 * 7 * 7 * 11 * 31 * 31 * 15107 * 66041 * 3629831 * 32303143 * 3644664791 * 3669739756759 * 9842283515159 * p34
3409719665791495912088557548714548870555928299445478456550560379913289306779291543643147189360470373 = 13 * 63079 * 1897177 * 6700089011507840929048948172217533113246337 * p45
1495013387078444830971209867702573522460009718985740283372631172961535442849970157536302235575256418 = 2 * 3 * 7 * 7 * 7 * 23 * 43 * 683 * 1069 * 11941 * 79943 * 174767 * 3055399 * 15229428191 * 97167102001 * 342494678174970979223 * p25
4525811937133892039004695583122389645196403182725701117694456147561445885523093800196418939098804213 = 3 * 7 * 127 * 181 * 971 * 1249 * 2417 * 938338889863 * 1435055134111 * 14576177493396886245107 * p39
2073311625512958451771092516352979522018668970964146344058841979417884026793493734398961719458991714 = 2 * 7 * 83 * 683 * 2927 * 2853337 * 14091887522033 * 229463148114318047593 * 119858213886351109516757 * p27
2898380564764331740650976346727898110985597681040329046593860520663948977367124731596264106185936093 = 3 * 3 * 101 * 151 * 271 * 1373 * 2153 * 17239 * 81239 * 2773871 * 7857481 * 899962939 * 82370557151 * 1455761853057791 * p28
3198477157946413712356473944979032360351567621787560078910437352396253167806937403734139451063110790 = 2 * 5 * 31 * 1543 * 31643 * 16930549 * 4315089697 * 147331735717257383607543819531715979 * p38
3122316024826623570999390780435061018523720250135414481535910292606995414401907413802207852764507150 = 2 * 5 * 5 * 13 * 53 * 9941 * 1095839 * 1436933 * 3566369 * 116278069 * 704550047 * 3739896647 * p46
1150279127692742245826975952975244459786635097720009368565532437170228458297750980752159689159559618 = 2 * 3 * 7 * 43 * 211 * 74843 * 487703 * 179282303 * 5191468757 * 279705792443006937347 * p45
3476994682280465721380107970080979643964773181753875578457560808630413498598886246958613562097879220 = 2 * 2 * 3 * 5 * 13 * 191 * 5912088495769122635780489 * 7782116779738216343902940458011407 * p36
2396930940518056525873755201487754594494952937746349126647824329949425055589025208870046787995076360 = 2 * 2 * 2 * 3 * 3 * 3 * 5 * 17 * 1361 * 13049 * 117243743 * 3537179857 * 13631427749 * 24865862748311267 * 83386179669921800083 * p24
5059444662280282391945087488109772319758228923707259715338382813207657103741168542070840844904146220 = 2 * 2 * 3 * 3 * 5 * 7 * 23 * 29 * 79 * 101 * 113 * 743 * 4831 * 2962207 * 4864957 * 15387939349 * 30166451450413001 * 22509974642730605057 * p23
9047865069205221785796363229531231728097227782231483365308238450563871586695628928251384364335276289 = 3 * 7 * 11 * 13 * 37 * 41 * 1327 * 1361 * 4243 * 29153 * 47513 * 9410983 * 55253941 * 219090281 * 1296987047 * 8269682713 * 84184045307 * p22
4818738698309238680979976982780739771434590565295500757251227197031479934194199257557409828463541162 = 2 * 31 * 79 * 379 * 25868233 * 64475011693 * 10929957278893 * 26357427070885165852945211371 * p34
7569920572151463447485708659596986502379885621035578545383014239308628711137659305904538890841093447 = 3 * 7 * 13 * 37 * 47 * 53 * 149 * 401 * 3457 * 34673 * 2494512199 * 13965069563 * 32420504751992087 * p44
7384425977124614373037190268769494322670515018022021984540726867123494391479847118318115045923964027 = 11 * 103 * 241 * 719 * 148709668309 * 51020820806647045557427500151889591 * p46
1460986172293537551768238256080060645982228305051951190204471703642679618463315789617586532311138454 = 2 * 3 * 31 * 47 * 673 * 1051 * 29478523 * 2049939233 * 120265998807826733 * 292426016421719911 * p39[/code]