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[ October 18, 2004 ]
Two Prime Numbers starting from 18 and 19.
by Farideh Firoozbakth (email).

Farideh Firoozbakht from Iran
found two nice probable primes  p1 and p2
formed by concatenating the integers
from 18 resp. 19 up to 3607 resp. 1471.

181920... ...36063607 (13296 digits)
and
192021... ...14701471 (4750 digits)

hitherto unknown in the prime number community.
Note that the endnumbers 1471 and 3607 of the strings
are primes, as well as 'their' concatenation 14713607 !

Farideh found these primes by using Mathematica that she
masters during 15 years now. "It is as necessary to me as air"
she writes in one of her messages.

Mathematica says p1 & p2 are primes, i.e.,

In[21]:= {PrimeQ[p1],PrimeQ[p2]}

Out[21]= {True, True}

Note that the built-in primality testing function PrimeQ
does not actually give a proof that a number is prime.
However, there are no known examples where PrimeQ fails.

In Mathematica we can generate p1 & p2 in this way :

p1=(v1={};Do[v1=Join[v1,IntegerDigits[k]],{k,18,3607}];FromDigits[v1])

p2=(v1={};Do[v1=Join[v1,IntegerDigits[k]],{k,19,1471}];FromDigits[v1])

And Mathematica determines length of p1 & p2 in this way:

In[24]:= Length[IntegerDigits[p1]]
Out[24]= 13296

In[25]:= Length[IntegerDigits[p2]]
Out[25]= 4750

Of course there is a limit with Mathematica but I think
the limit also depends on the computer that we use.
Mathematica can work on numbers with more than 30000000 digits.
For example:

In[20]:=
n=100000000;Timing[{Length[IntegerDigits[2^n]],DigitCount[2^n]}]

Out[20]=
{1185.36 Second,{30103000,{3009550,3011051,3005571,
3011482,3011589,3010173,3010805,3011875,3010874,3010030}}}

This means that Mathematica counts the digits of m = 2^100000000
and counts the digits 0 to 9 of m in 1185.36 seconds.
Indeed Mathematica says m is a 30103000-digit number and :

k — number of k in m

0 — 3009550
1 — 3011051
2 — 3005571
3 — 3011482
4 — 3011589
5 — 3010173
6 — 3010805
7 — 3011875
8 — 3010874
9 — 3010030

With the aid of the freeware tool PFGW ( PrimeForm)
I (PDG) could verify Farideh's results.
PFGW has a built-in function for Smarandache sequences 'Sm'.
Using the double coordinates Sm(x,y) makes for a circular
shift starting from y, so I had to divide the string with
the length of the shifted part. PFGW always rounds up
till an integer is left over. Reals cannot be input evidently.
Here are the commands and timings on my system
(Pentium 4 at 3 GHz) :


C:\PFGW>pfgw -q"sm(1471,19)/10^len(sm(18))"
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

sm(1471,19)/10^len(sm(18)) is 3-PRP! (1.5122s+0.0020s)


C:\PFGW>pfgw -q"sm(3607,18)/10^len(sm(17))"
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

sm(3607,18)/10^len(sm(17)) is 3-PRP! (15.6343s+0.0051s)


The lengths of the numbers can be determined as follows :


C:\PFGW>pfgw -od -f0 -q"len(sm(1471,19)/10^27)"
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

No factoring at all, not even trivial division
len(sm(1471,19)/10^27): 4750


C:\PFGW>pfgw -od -f0 -q"len(sm(3607,18)/10^25)"
PFGW Version 20040816.Win_Stable (v1.2 RC1a) [FFT v23.8]

No factoring at all, not even trivial division
len(sm(3607,18)/10^25): 13296


The Reference Table for
Smarandache Concatenated Numbers
with starting numbers from 1 to 19.
Maybe you like to participate ?

Start
from
Smarandache Concatenated Numbers
List of Probable Primesdigitlength
Searched up to
last term
1nihil1504164099
232, 98, 274410047
3192710067
474, 131410259
51721, 71129, 99185, 12325710351
672, 131210355
71311, 127267, 9492733, 717127571 PDG10001
[ 7171 dd. 25/10/2004 ]
892, 14933210135
918744510265
10nihil12701
1130980811041
1213410091
13729728066 PDG10265
[ 7297 dd. 23/10/2004 ]
14178, 476810089
15191010469
16435615677
17394610077
18360713296 FF11375
1914714750 FF10901


Further reading and exploring
Entries of F. Firoozbakht in Prime Curios!
Primes by Listing
Consecutive Number Sequences



A000160 Prime Curios! Prime Puzzle
Wikipedia 160 Le nombre 160














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E-mail address : pdg@worldofnumbers.com