The equation n!!(n-1)!!(n-2)!!(n-k)! is equal to y factorial finitely often. Some explicit examples are 5!!(4!!)(3!!)(2!!)(1!)= 6! , 3!!(2!!)(1!!)=3! , and 7!(6!!)(5!!)(4!!)(3!!)(2!!)(1!!)=10! Here the double factorial indicates the product of all odd or even integers less than or equal to that integer. For example, 5!!= 5*3*1 because 5, 3, and 1 are all the odd integers …
Continue reading “Conjecture no. 40 On super double factorials”
If 2+3+5…+n is equal to a prime , then it is also a prime of the form x^2+1 where n is greater than two finitely often? Some examples will follow 2+3= 2^2+1 2+3+5+7= 4^2+1 2+3+5+7+… 37 =14^2+1 It would be appreciated if someone could find more examples of this phenomenon. Regarding the composites of the …
x^y=y^x has two more solutions if extended to the Gaussian Integers. The two other solution of the equation x^y=y^x other than the trivial solutions (4,2) ,(2,4), (-2,4), (-4,2) is the Gaussian Integer pair (4+2i, 4+ 2i) and the conjugate (4-2i, 4-2i). It would be interesting to expand this identity over any 2^n Cayley-Dickson algebra with …
Continue reading “An interesting identity over the Gaussian Integers”
Is there any other Carmichael Numbers other than 1729 that is expressible as a compositorial number plus one i.e a compositorial number is the factorial of a number divided by the primoreal of the same number. If there is any other such examples they must by finite in number. https://oeis.org/A036691 https://en.wikipedia.org/wiki/Carmichael_number Thank you, Nunghead
Conjecture no. 34 There is finitely many highly composite numbers such that if one is added to ,it it is a perfect square. Conjecture no. 35 Is the square root of the highly composite numbers that satisfy conjecture no. 34 always prime if not is it prime infinitely often. https://en.m.wikipedia.org/wiki/Highly_composite_number https://oeis.org/A002182 Thank you, Nunghead
There always exists at least one prime in the interval, Ax and Bx when the GCD of a,b =1 where x is greater than 2. A and B also must be congruent modulo 2 I.e a-b must equal to an even number greater than zero ,a>b. https://en.m.wikipedia.org/wiki/Bertrand%27s_postulate
Is there arbitrarily long consecutive progressions of Gaussian Primes or any 2^n Cayley Dickson algebras. Thank you Nunghead
Does the sums of the reciprocals of figurate numbers equal to a transcendental number infinitely often. Also, if a regular polygon is not being able to be constructed by a straight edge and compass then does it mean that the pattern of the reciprocals of the objects forming the non constructable polygon summed is transcendental. …
A famous approximation is 22/7 nearly equal to 3.1428571 Another famous approximation is Plato’s approximation sqrt(2)+sqrt(3) nearly equal to 3.14626. Another approximation is the fourth root of 2143/22 given by Ramanujan. Some other approximations are sqrt(10) given by Aryabhatta and 355/113 given by some chinese mathematician. Some of my approximations are sqrt(9.9) correct to two …
Conjecture 27. Does 2+3+5+7+…n = x^2 have solutions finitely often. Here are some distinct solutions to this problem down below. Sum of prime numbers 2 to 23 = 100, a perfect square The next such sum is beyond the prime numbers under 20,000 Sum of prime numbers 2 to 22073 =25633969 Sq. root =5063 The …
Continue reading “Some conjectures on prime sums. Conjectures 27,28,29,30.”
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