By C. Ashbacher
Florentin Smarandache is actually a Renaissance guy, having produced caliber paintings in paintings, arithmetic and literature. this is often my 5th publication the place I extend on a few of his mathematical principles and there appears no lead to sight. In bankruptcy 1, numerous sequences created by way of concatenating the entire traditional numbers are tested. by means of expressing the numbers in bases except 10, a unmarried kind of concatenation can generate a number of diversified sequences. Upon exam, it may be obvious that there are a few major modifications among the sequences. Subsets of the usual numbers should be concatenated to create extra sequences and a number of other are tested in bankruptcy 2. The successive extraordinary numbers and the successive even numbers are concatenated to shape of the sequences which are thought of intimately. A stereogram is a two-dimensional picture that features a 3-dimensional photo. by means of it lengthy sufficient, the eyes lose concentration in exactly the proper manner and the picture seems to be. uncomplicated Smarandache stereograms may be developed utilizing simply characters and lots of various photographs are awarded 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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Example text
D1 represent element numbered n + 2, which is evenly divisible by 3. Form element number n + 3 by concatenating the next element K, where K is congruent to 1 modulo 3. This number can be represented as (dk . . d1 ) * 10r + K Since the product is evenly divisible by 3 and K is congruent to 1 modulo 3, the sum is congruent to 1 modulo 3. Form the next element by concatenating K + 1 to the previous one. This number can be represented in the form [ (dk . . d1 ) * 10r + K ]10s + (K+1) Since (dk . .
To investigate this question, the program was rewritten to determine what the cycle is for elements numbered greater than 10,000. That cycle turned out to be 82, 29, 58, 22, 51, 52. While the change in the cycles is understandable, given that numbers with an extra digit are being appended, there is no obvious explanation for the values of the cycle. Question: Are there any divisibility properties that can be used to predict the numbers in the division by 72 cycle for elements numbered greater than 100,000?
The program was then rerun searching for elements of the sequence that are divisible by 72 and the results for all elements with index less than or equal to 2000 are given in data listing 1. Data listing 1 For The For The For The term number 54, the number is divisible by 7 squared difference is 54 term number 146,the number is divisible by 7 squared difference is 92 term number 147, the number is divisible by 7 squared difference is 1 39 For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For The For term number 244, the number is divisible by 7 squared difference is 97 term number 245, the number is divisible by 7 squared difference is 1 term number 342, the number is divisible by 7 squared difference is 97 term number 343, the number is divisible by 7 squared difference is 1 term number 440, the number is divisible by 7 squared difference is 97 term number 441, the number is divisible by 7 squared difference is 1 term number 538, the number is divisible by 7 squared difference is 97 term number 539, the number is divisible by 7 squared difference is 1 term number 636, the number is divisible by 7 squared difference is 97 term number 637, the number is divisible by 7 squared difference is 1 term number 734, the number is divisible by 7 squared difference is 97 term number 735, the number is divisible by 7 squared difference is 1 term number 832, the number is divisible by 7 squared difference is 97 term number 833, the number is divisible by 7 squared difference is 1 term number 930, the number is divisible by 7 squared difference is 97 term number 931, the number is divisible by 7 squared difference is 1 term number 1043, the number is divisible by 7 squared difference is 112 term number 1059, the number is divisible by 7 squared difference is 16 term number 1084, the number is divisible by 7 squared difference is 25 term number 1190, the number is divisible by 7 squared difference is 106 term number 1206, the number is divisible by 7 squared difference is 16 term number 1231, the number is divisible by 7 squared difference is 25 term number 1337, the number is divisible by 7 squared difference is 106 term number 1353, the number is divisible by 7 squared 40 The For The For The For The For The For The For The For The For The For The For The For The For The For The difference is 16 term number 1378, difference is 25 term number 1484, difference is 106 term number 1500, difference is 16 term number 1525, difference is 25 term number 1631, difference is 106 term number 1647, difference is 16 term number 1672, difference is 25 term number 1778, difference is 106 term number 1794, difference is 16 term number 1819, difference is 25 term number 1925, difference is 106 term number 1941, difference is 16 term number 1966, difference is 25 the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared the number is divisible by 7 squared From this list, note that for elements numbered between 100 and 1000, the sequence of differences between elements divisible by 172 follow the 1, 97 cycle, and for the elements numbered greater than 1000, the difference cycle is 16, 25, 106.