Download Smarandache Sequences, Stereograms and Series by C. Ashbacher PDF

By C. Ashbacher

ISBN-10: 1931233233

ISBN-13: 9781931233231

Florentin Smarandache is really a Renaissance guy, having produced caliber paintings in paintings, arithmetic and literature. this can be my 5th booklet the place I extend on a few of his mathematical rules and there seems no lead to sight. In bankruptcy 1, numerous sequences created by means of concatenating all the ordinary numbers are tested. via expressing the numbers in bases except 10, a unmarried kind of concatenation can generate a number of varied sequences. Upon exam, it may be visible that there are a few major changes among the sequences. Subsets of the typical numbers could be concatenated to create extra sequences and a number of other are tested in bankruptcy 2. The successive peculiar numbers and the successive even numbers are concatenated to shape of the sequences which are thought of intimately. A stereogram is a two-dimensional photograph that includes a three-d photo. by means of taking a look at it lengthy adequate, the eyes lose concentration in exactly the correct means and the picture seems. basic Smarandache stereograms could be developed utilizing in simple terms characters and plenty of various photos are offered in bankruptcy three. The Smarandache services can be utilized together with different mathematical operations to create many various endless sequence. Many such sequence are tested in bankruptcy 4 and conclusions whether or not they converge or diverge are reached.

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The program to search the Prime Sequence for primes was modified to search for primes in the first 1000 terms of the above sequence. With the exception of the first element, no primes were found. Question: Is there an element in the sequence 3, 35, 357, 35711, . . other than the first that is prime? To some people, 1 is also a prime, so we can ask the similar question regarding the number of primes in the sequence 1, 12, 123, 1235, 12357, 1235711, 123571113, . . The program to search the Prime Sequence for primes was then modified to search for primes in the first 600 terms in the above sequence.

When the first 300 terms of this sequence are examined, ten primes are found and the element indices are 2, 3, 6, 8, 9, 15, 28, 57, 102 and 146. The relatively large number of elements that are prime raises interesting questions about what is different between this sequence and the others. To continue the examination of these sequences, the program was rewritten to examine the values of the terms modulo 7. It was used to count the numbers of elements of each of the three sequences 1, 12, 123, 1235, 12357, 1235711, .

Proof: A simple modulus 3 argument. Corollary: An element of the Reverse Sequence n(n-1) . . 321 can be prime only when n is congruent to 1 modulo 3. However, it is possible that there are more primes in the Reverse Sequence for bases other than 10, so the program was run to search for all primes in the first 500 elements of the Reverse Sequence for the bases 3 through 9. The results are summarized in table 4. Table 4 Base 3 4 5 6 7 8 9 Elements that are prime 2, 5, 13, 57 4, 106, 118, 130 2, 313 2, 6, 17, 28, 33, 37, 81 373 2, 9, 47, 50, 99 2, 5, 346 The final sequence described by Smarandache that uses all the positive integers is the Antisymmetric Sequence.

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