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## Department of Mathematics

## UNDERGRADUATE RESEARCH-MATH 444

Name: Mojtaba Moniri

Areas of Interest: Mathematical Logic, Number Theory

Description: One possibility is as follows. In Math 341 or elsewhere, you probably saw one-to-one correspondences between real numbers and sequences of natural numbers. Of the many ways a sequence of natural numbers could depend on a real parameter, this semester let us study a recent elaboration by Melvyn Nathanson in his May 2013 issue of the American Mathematical Monthly (see http://www.jstor.org/stable/10.4169/amer.math.monthly.120.05.409?origin=JSTOR-pdf ). From calculus, you remember that the roots, when the index goes to infinity, of a positive real number tend to 1. For a real number grater than 1 like r, isn't it appealing to you in order to isolate r more and more, to find the integer parts of the reciprocals of the errors corresponding to each root index. E.g., following this, the numbers 2 and e^(2/3), would correspond to the two sequences below (notice the pattern in the latter).

1 2 3 5 6 8 9 11 12 13 15 16 18 19 21

22 24 25 26 28 29 31 32 34 35 37 38 39 41 42

44 45 47 48 49 51 52 54 55 57 58 60 61 62 64

65 67 68 70 71 73 74 75 77 78 80 81 83 84 86

87 88 90 91 93 94 96 97 99 100 101 103 104 106 107

109 110 112 113 114 116 117 119 120 122 123 125 126 127 129

...

1 2 4 5 7 8 10 11 13 14 16 17 19 20 22

23 25 26 28 29 31 32 34 35 37 38 40 41 43 44

46 47 49 50 52 53 55 56 58 59 61 62 64 65 67

68 70 71 73 74 76 77 79 80 82 83 85 86 88 89

91 92 94 95 97 98 100 101 103 104 106 107 109 110 112

113 115 116 118 119 121 122 124 125 127 128 130 131 133 134

...

Learn, write up, and present the very readable mentioned article for 1 credit registration, or we can discuss the expectations for 2-credit registration (like having something to say on, as the author asks, how “almost periodic” the sequence of successive differences of the first sequence above is).