[1] Euclid (of Alexandria), “Elements, Book IX”, 300 BC.
[2] P.G.L. Dirichlet, “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält” Abhand. Ak. Wiss. Berlin, 1837.
[3] Neeraj Anant Pande, “Analysis of Primes Less than a Trillion”, International Journal of Computer Science & Engineering Technology (ISSN: 2229-3345), Vol. 6, No. 06, 2015, pp. 332 – 341.
[4] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 3n + k up to a Trillion”, IOSR Journal of Mathematics, Volume 11, Issue 3 Ver. IV, 2015, pp. 72-85.
[5] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 4n + k up to a Trillion”, International Journal of Mathematics and Computer Applications Research, Vol. 5, Issue 4, 2015, pp. 1-18.
[6] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 5n + k up to a Trillion”, Journal of Research in Applied Mathematics, Volume 2, Issue 5, 2015, pp. 14-29.
[7] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 6n + k up to a Trillion”, International Journal of Mathematics and Computer Research, Volume 3, Issue 6, 2015, pp. 1037-1053.
[8] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 7n + k up to a Trillion”, International Journal of Mathematics and Its Applications, Accepted, 2016.
[9] Neeraj Anant Pande, “Block-wise Distribution of Primes less than a Trillion in Arithmetical Progressions 8n + k”, IOSR Journal of Mathematics, Volume 12, Issue 3 Ver. V, 2016, PP 79-87.
[10] Neeraj Anant Pande, “Spacings Between and Units & Tens Place Digits in Primes till One Trillion in Arithmetical Progressions 8n + k”, American International Journal of Research in Science, Technology, Engineering and Mathematics, Issue 15, Volume 1, 2016, pp. 1–7.
[11] Neeraj Anant Pande, “Block-wise Density Distribution of Primes less than a Trillion in Arithmetical Progressions 9n + k”, International Journal of Advances in Mathematics and Statistics, Communicated, 2016.
[12] Neeraj Anant Pande, “Spacings Between and Units & Tens Place Digits in Primes till One Trillion in Arithmetical Progressions 9n + k”, International Journal of Mathematics and Statistics Invention, Volume 4, Issue 5, 2016, pp. 13–20.
[13] Neeraj Anant Pande, “Block-wise Density Distribution of Primes less than a Trillion in Arithmetical Progressions 10n + k”, Journal of Research in Applied Mathematics, Volume 2, Issue 8, 2016, pp. 10–18.
[14] Neeraj Anant Pande, “Evolution of Algorithms: A Case Study of Three Prime Generating Sieves”, Journal of Science and Arts, Year 13, No.3(24), 2013, pp. 267-276.
[15] Neeraj Anant Pande, “Algorithms of Three Prime Generating Sieves Improvised Through Nonprimality of Even Numbers (Except 2)”, International Journal of Emerging Technologies in Computational and Applied Sciences, Issue 6, Volume 4, 2013, pp. 274-279.
[16] Neeraj Anant Pande, “Algorithms of Three Prime Generating Sieves Improvised by Skipping Even Divisors (Except 2)”, American International Journal of Research in Formal, Applied & Natural Sciences, Issue 4, Volume 1, 2013, pp. 22-27.
[17] Neeraj Anant Pande, “Prime Generating Algorithms through Nonprimality of Even Numbers (Except 2) and by Skipping Even Divisors (Except 2)”, Journal of Natural Sciences, Vol. 2, No.1, 2014, pp. 107-116.
[18] Neeraj Anant Pande, “Prime Generating Algorithms by Skipping Composite Divisors”, International Journal of Computer Science & Engineering Technology, Vol. 5, No. 09, 2014, pp. 935-940.
[19] Neeraj Anant Pande, “Improved Prime Generating Algorithms by Skipping Composite Divisors and Even Numbers (Other Than 2)”, Journal of Science and Arts, Year 15, No.2(31), 2015, pp. 135-142.
[20] Neeraj Anant Pande, “Refinement of Prime Generating Algorithms”, International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 6, 2015, pp. 21-24.
[21] Herbert Schildt, “Java : The Complete Reference”, 7th Edition, Tata McGraw Hill, 2006.
[2] P.G.L. Dirichlet, “Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Differenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält” Abhand. Ak. Wiss. Berlin, 1837.
[3] Neeraj Anant Pande, “Analysis of Primes Less than a Trillion”, International Journal of Computer Science & Engineering Technology (ISSN: 2229-3345), Vol. 6, No. 06, 2015, pp. 332 – 341.
[4] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 3n + k up to a Trillion”, IOSR Journal of Mathematics, Volume 11, Issue 3 Ver. IV, 2015, pp. 72-85.
[5] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 4n + k up to a Trillion”, International Journal of Mathematics and Computer Applications Research, Vol. 5, Issue 4, 2015, pp. 1-18.
[6] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 5n + k up to a Trillion”, Journal of Research in Applied Mathematics, Volume 2, Issue 5, 2015, pp. 14-29.
[7] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 6n + k up to a Trillion”, International Journal of Mathematics and Computer Research, Volume 3, Issue 6, 2015, pp. 1037-1053.
[8] Neeraj Anant Pande, “Analysis of Primes in Arithmetical Progressions 7n + k up to a Trillion”, International Journal of Mathematics and Its Applications, Accepted, 2016.
[9] Neeraj Anant Pande, “Block-wise Distribution of Primes less than a Trillion in Arithmetical Progressions 8n + k”, IOSR Journal of Mathematics, Volume 12, Issue 3 Ver. V, 2016, PP 79-87.
[10] Neeraj Anant Pande, “Spacings Between and Units & Tens Place Digits in Primes till One Trillion in Arithmetical Progressions 8n + k”, American International Journal of Research in Science, Technology, Engineering and Mathematics, Issue 15, Volume 1, 2016, pp. 1–7.
[11] Neeraj Anant Pande, “Block-wise Density Distribution of Primes less than a Trillion in Arithmetical Progressions 9n + k”, International Journal of Advances in Mathematics and Statistics, Communicated, 2016.
[12] Neeraj Anant Pande, “Spacings Between and Units & Tens Place Digits in Primes till One Trillion in Arithmetical Progressions 9n + k”, International Journal of Mathematics and Statistics Invention, Volume 4, Issue 5, 2016, pp. 13–20.
[13] Neeraj Anant Pande, “Block-wise Density Distribution of Primes less than a Trillion in Arithmetical Progressions 10n + k”, Journal of Research in Applied Mathematics, Volume 2, Issue 8, 2016, pp. 10–18.
[14] Neeraj Anant Pande, “Evolution of Algorithms: A Case Study of Three Prime Generating Sieves”, Journal of Science and Arts, Year 13, No.3(24), 2013, pp. 267-276.
[15] Neeraj Anant Pande, “Algorithms of Three Prime Generating Sieves Improvised Through Nonprimality of Even Numbers (Except 2)”, International Journal of Emerging Technologies in Computational and Applied Sciences, Issue 6, Volume 4, 2013, pp. 274-279.
[16] Neeraj Anant Pande, “Algorithms of Three Prime Generating Sieves Improvised by Skipping Even Divisors (Except 2)”, American International Journal of Research in Formal, Applied & Natural Sciences, Issue 4, Volume 1, 2013, pp. 22-27.
[17] Neeraj Anant Pande, “Prime Generating Algorithms through Nonprimality of Even Numbers (Except 2) and by Skipping Even Divisors (Except 2)”, Journal of Natural Sciences, Vol. 2, No.1, 2014, pp. 107-116.
[18] Neeraj Anant Pande, “Prime Generating Algorithms by Skipping Composite Divisors”, International Journal of Computer Science & Engineering Technology, Vol. 5, No. 09, 2014, pp. 935-940.
[19] Neeraj Anant Pande, “Improved Prime Generating Algorithms by Skipping Composite Divisors and Even Numbers (Other Than 2)”, Journal of Science and Arts, Year 15, No.2(31), 2015, pp. 135-142.
[20] Neeraj Anant Pande, “Refinement of Prime Generating Algorithms”, International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 6, 2015, pp. 21-24.
[21] Herbert Schildt, “Java : The Complete Reference”, 7th Edition, Tata McGraw Hill, 2006.
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Affiliations
Neeraj Anant Pande
Affiliation not stated
How to Cite
Pande, N. (2016, July 26). Spacings Between And Units & Tens Place Digits In Primes Till One Trillion In Arithmetical Progressions 10n + K. International Journal of Engineering Maths and Computer Science, 4(7). https://doi.org/https://doi.org/10.15520/.2016.vol4.iss7.11
Spacings Between And Units & Tens Place Digits In Primes Till One Trillion In Arithmetical Progressions 10n + K
Abstract
Prime numbers in arithmetical progressions 10n + k are analyzed for the spacings between primes of same form in blocks of 10n till 1012. The minimum spacings between primes in same progressions, the first and last prime starters with minimum spacings, and also the frequency of prime pairs with minimum spacings within blocks of 10n are determined. Similar analysis for maximum spacings between primes of same form in blocks of 10n till 1012 is carried out. The digits in units place and digits in units & tens place in primes of these forms are considered for abundance.