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Exploring Prime Distances With MATHEMATICA and The VOYAGE 200 CAS

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Abstract:

The fascinating dichotomy with regards to the prime numbers in the sense that their infinitude on the one hand is offset by the ability to create prime gaps as large as one pleases in no small measure renders their appeal. An intriguing aspect associated with the primes is the distance between a positive integer and the nearest prime, called the prime distance function. The prime distance $pd(n)$ for a non-negative integer n is the absolute difference between n and the nearest prime. One may round up or down in case of ties. To cite an example, $pd(93) =4$; for the nearest prime to 93 is 97 if we round up or 89 if we round down. I choose to round up in such cases. Associated with the prime distance function are prime gaps of length n which constitutes a run of n - 1 consecutive composite integers between two successive primes. In the above example, one has a gap of 8 between the primes 89 and 97 and hence a run of 7 consecutive composites. In this paper, we will construct a scatter plot for the prime distance function and examine several associated integer sequences. Our culminating activity will include simple programs for The VOYAGE 200 to obtain the nearest prime and the previous prime. A program already exists in the manual to obtain the Next Prime. MATHEMATICA 5.2 in its Number Theory Add-On Package has commands for the Next Prime and the Previous Prime. Please join us to witness CAS technology furnishing new and exciting explorations related to the remarkable prime numbers.
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Name: Mathematical Association of America
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MLA Citation:

SCHIFFMAN, JAY. "Exploring Prime Distances With MATHEMATICA and The VOYAGE 200 CAS" Paper presented at the annual meeting of the Mathematical Association of America, The Fairmont Hotel, San Jose, CA, Aug 03, 2007 <Not Available>. 2009-05-24 <http://www.allacademic.com/meta/p204466_index.html>

APA Citation:

SCHIFFMAN, J. L. , 2007-08-03 "Exploring Prime Distances With MATHEMATICA and The VOYAGE 200 CAS" Paper presented at the annual meeting of the Mathematical Association of America, The Fairmont Hotel, San Jose, CA <Not Available>. 2009-05-24 from http://www.allacademic.com/meta/p204466_index.html

Publication Type: Conference Paper/Unpublished Manuscript
Abstract: The fascinating dichotomy with regards to the prime numbers in the sense that their infinitude on the one hand is offset by the ability to create prime gaps as large as one pleases in no small measure renders their appeal. An intriguing aspect associated with the primes is the distance between a positive integer and the nearest prime, called the prime distance function. The prime distance $pd(n)$ for a non-negative integer n is the absolute difference between n and the nearest prime. One may round up or down in case of ties. To cite an example, $pd(93) =4$; for the nearest prime to 93 is 97 if we round up or 89 if we round down. I choose to round up in such cases. Associated with the prime distance function are prime gaps of length n which constitutes a run of n - 1 consecutive composite integers between two successive primes. In the above example, one has a gap of 8 between the primes 89 and 97 and hence a run of 7 consecutive composites. In this paper, we will construct a scatter plot for the prime distance function and examine several associated integer sequences. Our culminating activity will include simple programs for The VOYAGE 200 to obtain the nearest prime and the previous prime. A program already exists in the manual to obtain the Next Prime. MATHEMATICA 5.2 in its Number Theory Add-On Package has commands for the Next Prime and the Previous Prime. Please join us to witness CAS technology furnishing new and exciting explorations related to the remarkable prime numbers.

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