[ *October 18, 2004* ]

Two Prime Numbers starting from 18 and 19

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 Primes_{digitlength} | Searched up to last term |

1 | nihil | 15041_{64099} |

2 | 3_{2}, 9_{8}, 27_{44} | 10047 |

3 | 19_{27} | 10067 |

4 | 7_{4}, 13_{14} | 10259 |

5 | 17_{21}, 71_{129}, 99_{185}, 123_{257} | 10351 |

6 | 7_{2}, 13_{12} | 10355 |

7 | 13_{11}, 127_{267}, 949_{2733}, 7171_{27571 PDG} | 10001 [ 7171 dd. 25/10/2004 ] |

8 | 9_{2}, 149_{332} | 10135 |

9 | 187_{445} | 10265 |

10 | nihil | 12701 |

11 | 309_{808} | 11041 |

12 | 13_{4} | 10091 |

13 | 7297_{28066 PDG} | 10265 [ 7297 dd. 23/10/2004 ] |

14 | 17_{8}, 47_{68} | 10089 |

15 | 19_{10} | 10469 |

16 | 43_{56} | 15677 |

17 | 39_{46} | 10077 |

18 | 3607_{13296 FF} | 11375 |

19 | 1471_{4750 FF} | 10901 |

Further reading and exploring

Entries of F. Firoozbakht in Prime Curios!

Primes by Listing

Consecutive Number Sequences