Problem 45994. Investigate the frequency of last digits of prime numbers

Created by ChrisR

Solve Later

{"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","targetMode":"","relationshipId":"rId1","target":"/matlab/document.xml"},{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/output","targetMode":"","relationshipId":"rId2","target":"/matlab/output.xml"}],"parts":[{"partUri":"/matlab/document.xml","relationship":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>The last digit of a prime number greater than 5 can be 1, 3, 7, or 9. If the primes are distributed randomly, then these digits should be equally likely. However,</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html\"><w:r><w:t>mathematicians discovered</w:t></w:r></w:hyperlink><w:r><w:t> relatively recently that--as Robert Lemke Oliver put it--these digits \"really hate to repeat themselves\". In other words, last digits repeat less often than expected.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91), and none of them are followed by a prime ending in a 1. In fact, none of the primes less than 100 and ending in 3, 7, or 9 are followed by a prime ending in 3, 7, or 9, respectively.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Write a function to compute the frequency of the last digits of primes between 7 and the input number. Return a matrix whose rows correspond to the digits of the first prime and columns correspond to the digits of the next prime. Please remember to (a) omit 2, 3, and 5 and (b) account for the prime following the last prime in your list. For the example given above, the function should return</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ 0 0.6000 0.4000 0\n 0 0 0.3333 0.6667\n 0.6667 0.1667 0 0.1667\n 0.4000 0.4000 0.2000 0]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>For example, six primes less than 100 end in 7. They are followed by four primes ending in 1 (including 101), one prime ending in 3, zero primes ending in 7, and one prime ending 9. The frequencies, reported in the third row, are thus 2/3, 1/6, and 1/6.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

The last digit of a prime number greater than 5 can be 1, 3, 7, or 9. If the primes are distributed randomly, then these digits should be equally likely. However, <a href = "https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html">mathematicians discovered</a> relatively recently that--as Robert Lemke Oliver put it--these digits "really hate to repeat themselves". In other words, last digits repeat less often than expected.For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91), and none of them are followed by a prime ending in a 1. In fact, none of the primes less than 100 and ending in 3, 7, or 9 are followed by a prime ending in 3, 7, or 9, respectively.Write a function to compute the frequency of the last digits of primes between 7 and the input number. Return a matrix whose rows correspond to the digits of the first prime and columns correspond to the digits of the next prime. Please remember to (a) omit 2, 3, and 5 and (b) account for the prime following the last prime in your list. For the example given above, the function should return<pre> 0 0.6000 0.4000 0
 0 0 0.3333 0.6667
 0.6667 0.1667 0 0.1667
 0.4000 0.4000 0.2000 0</pre>For example, six primes less than 100 end in 7. They are followed by four primes ending in 1 (including 101), one prime ending in 3, zero primes ending in 7, and one prime ending 9. The frequencies, reported in the third row, are thus 2/3, 1/6, and 1/6.

The last digit of a prime number greater than 5 can be 1, 3, 7, or 9. If the primes are distributed randomly, then these digits should be equally likely. However, <https://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html mathematicians discovered> relatively recently that--as Robert Lemke Oliver put it--these digits "really hate to repeat themselves". In other words, last digits repeat less often than expected.

For example, five primes less than 100 end in a 1 (11, 31, 61, 71, and 91), and none of them are followed by a prime ending in a 1. In fact, none of the primes less than 100 and ending in 3, 7, or 9 are followed by a prime ending in 3, 7, or 9, respectively.

Write a function to compute the frequency of the last digits of primes between 7 and the input number. Return a matrix whose rows correspond to the digits of the first prime and columns correspond to the digits of the next prime. Please remember to (a) omit 2, 3, and 5 and (b) account for the prime following the last prime in your list. For the example given above, the function should return

0    0.6000    0.4000         0
         0         0    0.3333    0.6667
    0.6667    0.1667         0    0.1667
    0.4000    0.4000    0.2000         0

For example, six primes less than 100 end in 7. They are followed by four primes ending in 1 (including 101), one prime ending in 3, zero primes ending in 7, and one prime ending 9. The frequencies, reported in the third row, are thus 2/3, 1/6, and 1/6.

Solve

Solution Stats

20 Solutions
10 Solvers

Last Solution submitted on Jul 26, 2021

Last 200 Solutions

Problem Comments

Problem Recent Solvers10

Problem Tags

number theory primes

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!

Problem 45994. Investigate the frequency of last digits of prime numbers

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers10

Suggested Problems

More from this Author131

Problem Tags

Community Treasure Hunt

Problem 45994. Investigate the frequency of last digits of prime numbers

Solution Stats

Last 200 Solutions

Problem Comments

Problem Recent Solvers10

Suggested Problems

More from this Author131

Problem Tags

Community Treasure Hunt

Players