interlaced primes

davar55 · 2013-12-30, 04:19

An "interlaced prime" is a prime number such that
the two integers obtained by taking all odd indexed
digits and all even indexed digits are also prime.

For example,

3779 is an interlaced prime, because 37 and 79 are prime.
233 is one, because 23 and 3 are.
1239 isn't one, because it's not prime, though 13 and 29 are.

The puzzle is: find a 100, 200, and 300 digit interlaced prime.

Batalov · 2013-12-30, 05:31

Too easy for 100-digit primes:

Code:

11*10^98+1541717
11*10^98+4531811
11*10^98+106066899
11*10^98+106445519
11*10^98+106521611
11*10^98+106541717
11*10^98+107551019
11*10^98+109231679
11*10^98+204521411
11*10^98+205495211
11*10^98+303281273
11*10^98+401531819
11*10^98+402231779
11*10^98+403521911
11*10^98+405551519
11*10^98+500030199
11*10^98+502475519
11*10^98+601211471
11*10^98+602010393
11*10^98+609036291
11*10^98+703086993
11*10^98+709271273
11*10^98+803231873
...and many more

Batalov · 2013-12-30, 05:41

Too easy for 200-digits...

Code:

11*10^198+72196473
11*10^198+87700913
11*10^198+93684491
11*10^198+1007993033
11*10^198+1009744239
11*10^198+1017145719
11*10^198+1017469191
11*10^198+1057100093
11*10^198+1073556197
11*10^198+1121576013
11*10^198+1160297333
11*10^198+1170170433
11*10^198+1176140637
11*10^198+1180393373
11*10^198+1198964277
11*10^198+2108601377
...etc etc

axn · 2013-12-30, 06:01

300 digits - random numbers

Code:

293986740354515974790582626614765781963790658971301820571570509792645132253746095100028543495297221501838942920800632945888406431455467662980448856097
137924486521064589360173063150586089185092718642783171287032769462279303511197175016632566802862417120885776980651782057713542310483708381166419323547
219337998264744806355241501654957849739600518723602663611540756856708819916835709902675188967412370813812701527817507302570699749622624759133023255131714967019755100106062382554636489052289672242117510210883885974726992800860501673822904557878183450462433110445853476078636821918606444189835263059477

Batalov · 2013-12-30, 06:02

Too easy for 300-digits...

Code:

11*10^298+70879871
11*10^298+1144875071
11*10^298+1153537431
11*10^298+1194554993
11*10^298+2036026719

Too easy for 2000-digits...
11*10^1998+8520244037

axn · 2013-12-30, 06:05

400 digits

Code:

8350751504598260235458208041132589283895108593301912957596812058410179150271323816926692092563569856\
7907921594393698305085626574476809962044593679891704157893151740857982911692361267127363151545819241

1294576894946799337882630471201911143523625051893816911882551430225228207512280350988995174218260463\
6314421625209204133966253756720033657129514452996786483283432357369345056790420965306475912377324077

8132590475571658094459948627690923335748588226038004471112302159819121843385925316028550953138091398\
1126995171589862851521045380421205127298125007257112322830831560992868699925019724526138526690845663\
7693017494221156924532903962908431035309865662256357754647726080039396652701424955913464759289991677\
0846145873829833145312734507835679938425901516679920346210296675132076346735195112534757831294204717

Batalov · 2013-12-30, 06:11

...and 11*10^1998+106032915291

On to 5000-digit numbers, then, eh? Or straight to 10000-digits?

axn · 2013-12-30, 06:26

Quote:

Originally Posted by Batalov

...and 11*10^1998+106032915291

On to 5000-digit numbers, then, eh? Or straight to 10000-digits?

I vote straight to 10000

How are you generating these numbers? By my estimate, if you have about 100 random primes of half size with the same (mod 3), then with the 100*99 combinations you get, should yield about 1-2 full size primes.

Batalov · 2013-12-30, 06:34

You got it. Get some half size numbers, interlace them with a perl script and pfgw. With 10,000-digits, one should also prove the 10,000-digit prime. ;-)
Not that it would be too hard. Piece of cake.

Here's a 3,000-digit one: 11*10^2998+5262550199

axn · 2013-12-30, 06:43

1000 digit - random numbers. Last one is not proven.

Code:

3668350082303612721273492182800265445256942323251842505487416615983292096421756092607251946695761683\
8245431656406765864772445957158119960843071438283411570630711430962493485126327819869590469603712150\
8551635549402225851396975828310280987207283318794540794639324669596229937291990801449166358038568729\
6249765336275078581664967430901264828142366921667770784348737001648317295398774101378033869442117131\
9730742324925611759318648289002016906037090363477626149373093371795820082586312006447800516698569551

9603619310404878826169079490144507491058266294675582784215093981519487091514526157165500831856966121\
0060536210496076195145217315962484713695594795636547337698113102738264866786444849658031729383806736\
9212334197929366465742347809266891493568724679275134728945442313830441995004676793892930582864098584\
5880125925308193022268542476528463372294159654493726130581258222255695268119996304259828468302724935\
1466275365800090699459099822096911982910350398023678489955021267060523786664880057250681185680646553

3966608336510903812034003468172878221621763940972914892081040425605744495120556892462632293426571585\
4822570854428175401963691851958139249827009961452114755266019527610675255010984361689556796616618231\
8020465045331662516044096670675681694571742542415793517519568214189497610386493505791447398526833645\
1417537307663908711113413002976328429634488656172866342474881499866598509301476299630833781026175306\
8952511263335451499749022923265684561537946293745788208932160628890194897325067827823436178972974551\
4304779248693495342442636193589360242491993975209014969706870913484992196360355882083684506988752894\
6528489071625539326523705801798350821262664895647244370695021824664383278212492431656996251464697377\
7206718340354881723578020212624585361975229658319189797946130014327589083238846698434022171274193315\
9174360672472533264598205060191076599934158960498928829200092609116199086209317003950033693840727366\
2768144899397535009231327617709650852203078826568664381820000567424570860801511865669880566496555513

Batalov · 2013-12-30, 07:35

And a 5000-digit prime: 11*10^4998+231062857013
(two 2500s: 10^2499+216871 and 10^2499+302503)

And a 10,000-digit (PR)prime: 11*10^9998+1354541841513 (maybe will prove, maybe not ;-)
(The interlaced 5000s: 10^4999+344811 and 10^4999+1551453)

...and 11*10^9998+10600811061999, to boot.

2013-12-30, 04:19	#1
davar55 May 2004 New York City 108A₁₆ Posts	interlaced primes An "interlaced prime" is a prime number such that the two integers obtained by taking all odd indexed digits and all even indexed digits are also prime. For example, 3779 is an interlaced prime, because 37 and 79 are prime. 233 is one, because 23 and 3 are. 1239 isn't one, because it's not prime, though 13 and 29 are. The puzzle is: find a 100, 200, and 300 digit interlaced prime. Last fiddled with by davar55 on 2013-12-30 at 05:01

2013-12-30, 07:35	#11
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶×13 Posts	And a 5000-digit prime: 1110^4998+231062857013 (two 2500s: 10^2499+216871 and 10^2499+302503) And a 10,000-digit (PR)prime: 1110^9998+1354541841513 (maybe will prove, maybe not ;-) (The interlaced 5000s: 10^4999+344811 and 10^4999+1551453) ...and 1110^9998+10600811061999, to boot. Last fiddled with by Batalov on 2013-12-30 at 10:08 Reason: added two 10,000-digits*

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2013-12-30, 06:02	#5
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶·13 Posts	Too easy for 300-digits... Code: 1110^298+70879871 1110^298+1144875071 1110^298+1153537431 1110^298+1194554993 1110^298+2036026719 Too easy for 2000-digits... 1110^1998+8520244037

2013-12-30, 06:11	#7
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶×13 Posts	...and 11*10^1998+106032915291 On to 5000-digit numbers, then, eh? Or straight to 10000-digits?

2013-12-30, 06:34	#9
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶×13 Posts	You got it. Get some half size numbers, interlace them with a perl script and pfgw. With 10,000-digits, one should also prove the 10,000-digit prime. ;-) Not that it would be too hard. Piece of cake. Here's a 3,000-digit one: 11*10^2998+5262550199