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            <title xml:lang="en">Prime Numbers Networks: Visualizing and Characterizing Relationships Between Prime Numbers</title>
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                <forename type="first">Luciano</forename>
                <surname>da Fontoura Costa</surname>
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                <forename>Luciano</forename>
                <surname>Da F. Costa</surname>
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                <title xml:lang="en">Prime Numbers Networks: Visualizing and Characterizing Relationships Between Prime Numbers</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Luciano</forename>
                    <surname>da Fontoura Costa</surname>
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              <p>Prime numbers constitute mathematical concepts of particular interest as a consequence of the their relative paucity among integer values, as well as to the relative unpredictability of their appearance and density along the sequence of integer values. As a consequence of their own definition, prime numbers can be though as being relatively uncommon values along the integers. However, at the same time, prime numbers constitute the building blocks of all the other, composite, numbers, which gives rise to the interesting and important concept of prime factor decomposition. While this concept provides an interesting approach for quantifying interrelationships between composite numbers, it cannot be applied to pairs of prime numbers, as they do not share any factor distinct from 1. In this work, the sets of shortest paths between the prime numbers in respective prime factors networks are taken into account as the primary subsidy for estimating, in quantitative manner, interrelationships between pairs of prime numbers. More specifically, we take into account not only the length and number of shortest paths between each pair of prime numbers in a respective prime factor network, but also quantify the independence (which is related to the entanglement) between the shortest paths in a same given set. This is performed in terms of the concept of effective widths along the hierarchical levels of a set of shortest paths, corresponding to the exponential entropy of the respective sets of transition probabilities. An alternative approach, in which the shared links have their transition probabilities added, is also described for defining effective widths. Several interesting results are reported and discussed, including the identification of intrinsic heterogeneity of interconnections in prime numbers networks and the marked diversity of independence between the shortest paths in the sets respective to pairs of prime numbers. These interrelationships between shortest paths arequantified in terms of statistics of the respective effective width and efficiency index, resulting in relatively intricate distributions among the considered pairs of prime numbers. In particular, several of these indices reach the largest values along an "L"-shaped crest.</p>
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