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Old 2022-03-30, 23:14   #1
sweety439
 
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

D8916 Posts
Default 432

432 is the only number k besides -1 (which corresponds to Catalan conjecture) such that the Mordell curves x^2+k=y^3 (which are specific elliptic curves) has rational solutions with both x and y nonzero, but only finitely many rational solutions (only consider primitive k, i.e. sixth-power-free k).

432 is probably the only non-Pronic number >1 which is not the sum of a Pronic number and k*p, where k is 1 or 2 and p is prime

432+-1 are both primes
(432+-2)/2 are both semiprimes (note that they are sum of cubes and difference of cubes, thus cannot be primes and can only be semiprimes)
(432+-3)/3 are both semiprimes
(432+-4)/4 are both primes
432+-5 are both semiprimes
(432+-6)/6 are both primes

432/2 = 216, the Plato number
216 = 6^3 (the smallest cube which is not prime power), and 216+-1, 216+-2, 216+-3 are semiprimes

432/3 = 144

(144-d)/d is prime for all divisors d of 12 except 1 and 4 (note that this number for d = 1 and 4 cannot be primes, because of algebra factors: 144-1 = (12-1) * (12+1), and 144-4 = (12-2) * (12+2), and can only be semiprimes)
144+-1 are both squarefree semiprimes
(144+-2)/2 are both primes
(144+3)/3 is square of prime, (144-3)/3 is prime
(144+4)/4 is prime, (144-4)/4 cannot be prime since it is difference of square (both 144 and 4 are squares) and can only be semiprime
144+-5 are both primes
(144+6)/6 is square of prime, (144-6)/6 is prime
144+-7 are both primes
(144+-8)/8 are both primes
(144+9)/9 is prime, (144-9)/9 cannot be prime since it is difference of square (both 144 and 9 are squares) and can only be semiprime
144+-11 are both squarefree semiprimes
(144+-12)/12 are both primes
144+-13 are both primes
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