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Numbers and Elementary Mathematics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Twin primes are prime numbers that differ by two: (3,5) $ (3, 5) $ , (5,7) $ (5, 7) $ , (11,13) $ (11, 13) $ , (17,19) $ (17, 19) $ , . . . . It is not known whether there are infinitely many twin primes. The sum of the reciprocals of the twin primes converges; the value
Fostering collateral creativity through teaching school mathematics with technology: what do teachers need to know?
Published in International Journal of Mathematical Education in Science and Technology, 2022
Wolfram Alpha, as an instant provider of information, generates several prime numbers greater than 2021 (Figure 9). Among them are 2027 and 2029. One can (or has to be prompted to) note that the two prime numbers are special because they differ by 2. Such primes are called twin primes. Figure 9 displays the following pairs of twin primes: (2027, 2029), (2081, 2083), (2087, 2089), (2111, 2113), (2129, 2131), (2141, 2143). One may observe (Figure 9) that among 20 prime numbers more than half are twin primes. Although it is known from the time of Euclid that the number of primes is infinite, it is not known, as of the time of writing this paper, whether there are infinitely many twin primes. In the spirit of Dewey’s (1933) reflective inquiry as integration of knowing and doing and Piaget’s (1973) reflective abstraction as creative reconstruction of previous experience, the questions about twin primes, with answers easily provided by Wolfram Alpha as the nucleus of Vygotsky’s (1930) instrumental act, may continue as follows. Can one describe pairs of twin primes based on their last digits? Why may not other digits serve as the last digits of twin primes?Consider the pair (2027, 2029). What is the next pair of twin primes with 7 and 9 as the last digits?Consider the pair (2081, 2083). What is the next pair of twin primes with 1 and 3 as the last digits?Consider the pair (2129, 2131). What is the next pair of twin primes with 9 and 1 as the last digits?What is the difference between 2027 and the next twin prime with the last digit 7?What is the difference between 2081 and the next twin prime with the last digit 1?What is the difference between 2129 and the next twin prime with the last digit 9?