Raphie Frank
Perfect 31, Perfect 137, Mersenne Primes and Perfect Numbers
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From Wikipedia on the number 31
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Thirty-one is the third Mersenne prime ( 2^5 - 1 ) as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 2^5 - 1 ( 2^5 - 1). 31 is the eighth Mersenne prime exponent. 31 is also the fourth lucky prime.
31 is a centered triangular number, a centered pentagonal number and centered decagonal number. At 31, the Mertens function sets a new low of -4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number.
31 is a repdigit in base 5 (111), and base 2 (11111).
The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; the factorisations of the first two are:
* 333333331 = 17*19607843
* 3333333331 = 673*4952947
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from Wikipedia on the number 137...
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One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n − 1. It is also the fourth Stern prime. 137 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.
Using two radii to divide a circle according to the golden ratio yields sectors of approximately 137° (the golden angle) and 222°...
In physics... the numerical value of the fine structure constant is almost precisely 1/137. This quantity is given by combining several fundamental constants of nature...
... the importance of the constant is that it measures the strength of the electromagnetic interaction. It is precisely because the constant is so small (i.e. 1/137 as opposed to 1/3 or 5 or 100...) that quantum electrodynamics (QED) works so amazingly well as a quantum theory of electromagnetism
Perfect 31, Perfect 137, Mersenne Primes and Perfect Numbers
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From Wikipedia on the number 31
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Thirty-one is the third Mersenne prime ( 2^5 - 1 ) as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496, since 496 = 2^5 - 1 ( 2^5 - 1). 31 is the eighth Mersenne prime exponent. 31 is also the fourth lucky prime.
31 is a centered triangular number, a centered pentagonal number and centered decagonal number. At 31, the Mertens function sets a new low of -4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number.
31 is a repdigit in base 5 (111), and base 2 (11111).
The numbers 31, 331, 3331, 33331, 333331, 3333331, and 33333331 are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; the factorisations of the first two are:
* 333333331 = 17*19607843
* 3333333331 = 673*4952947
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from Wikipedia on the number 137...
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One hundred [and] thirty-seven is the 33rd prime number; the next is 139, with which it comprises a twin prime, and thus 137 is a Chen prime. 137 is an Eisenstein prime with no imaginary part and a real part of the form 3n − 1. It is also the fourth Stern prime. 137 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes.
Using two radii to divide a circle according to the golden ratio yields sectors of approximately 137° (the golden angle) and 222°...
In physics... the numerical value of the fine structure constant is almost precisely 1/137. This quantity is given by combining several fundamental constants of nature...
... the importance of the constant is that it measures the strength of the electromagnetic interaction. It is precisely because the constant is so small (i.e. 1/137 as opposed to 1/3 or 5 or 100...) that quantum electrodynamics (QED) works so amazingly well as a quantum theory of electromagnetism