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SOME NOTIONS AND QUESTIONS IN NUMBER THEORY Vol. II

IP.com Disclosure Number: IPCOM000128801D
Original Publication Date: 1999-Sep-11
Included in the Prior Art Database: 2005-Sep-19
Document File: 18 page(s) / 42K

Publishing Venue

Software Patent Institute

Related People

Mihaly Bencze: AUTHOR [+4]

Abstract

The second volume of Smarandache sequences, following that edited by the late C. Dumitrescu and V. Seleacu from the University of Craiova, is been published by the American Research Press.

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 12% of the total text.

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

Copyright American Research Press, 1999 All Rights Reserved

SOME NOTIONS AND QUESTIONS IN NUMBER THEORY Vol. II

edited by

Mihaly Bencze Lucian Tutescu 6, Harmanului Street Mathematics Department 2212 Sacele 3 Fratii Buzesti College Jud. Brasov, Romania Craiova, R-1100, Romania

The second volume of Smarandache sequences, following that edited by the late C. Dumitrescu and V. Seleacu from the University of Craiova, is been published by the American Research Press.

We want to thank Dr. M. L. Perez and J. Castillo for their help and encouragements in this action.

1) Smarandache Concatenate Natural Sequence:

1,22,333,4444,55555,666666,7777777,88888888,999999999, 10101010101010101010,1111111111111111111111, 121212121212121212121212,13131313131313131313131313, 1414141414141414141414141414,151515151515151515151515151515,...

2) Smarandache Concatenated Prime Sequence:

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, ...

Smarandache Back Concatenated Prime Sequence: 2, 32, 532, 7532, 117532, 13117532, 1713117532, 191713117532, 23191713117532, ...

Conjecture: there are infinitely many primes among the first sequence numbers!

3) Smarandache Concatenated Odd Sequence:

1, 13, 135, 1357, 13579, 1357911, 135791113, 13579111315, 1357911131517, ...

Smarandache Back Concatenated Odd Sequence: 1, 31, 531, 7531, 97531, 1197531, 131197531, 15131197531, 1715131197531, ...

Conjecture: there are infinitely many primes among these numbers!

4) Smarandache Concatenated Even Sequence:

Mihaly Bencze Lucian Tutescu Page 1 Sep 11, 1999

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SOME NOTIONS AND QUESTIONS IN NUMBER THEORY Vol. II

2, 24, 246, 2468, 246810, 24681012, 2468101214, 246810121416, ...

Smarandache Back Concatenated Even sequence: 2, 42, 642, 8642, 108642, 12108642, 1412108642, 161412108642, ...

Conjecture: none of them is a perfect power!

5) Smarandache Concatenated S-Sequence {generalization}:

Let s1, s2, s3, s4, ..., sn, ... be an infinite integer sequence (noted by S).

Then: ____ ______ ________ _____________ s1, s1s2, s1s2s3, s1s2s3s4, s1s2s3s4...sn, ... is
called the Concatenated S-sequence, ____ ______ ________ _____________ s1, s2s2,
s3s2s1, s4s3s2s1, sn...s4s3s2s1, ... is called the Back Concatenated S-sequence.

Questions: a) How many terms of the Concatenated S-sequence belong to the initial S- sequence? b) Or, how many terms of the Concatenated S-sequence verify the relation of other given sequences?

The first three cases are particular.

Look now at some other examples, when S is the sequence of squares, cubes, Fibonacci respectively (and one can go so on):

Smarandache Concatenated Square Sequence: 1, 14, 149, 14916, 1491625, 149162536, 14916253649, 1491625364964, ...

Smarandache Back Concatenated Square Sequence: 1, 41, 941, 16941, 2516941, 362516941, 49362516941, 6449362516941, ...

How many of them are perfect squares?

Smarandache Concatenated Cubic Sequence: 1, 18, 1827, 182764, 1...