DULA

It’s brb.io OK? #[1]​ #relays #nostr

#[0]​ ₿ ! Where is the Bitcoin emoji? Why this is not out there yet? #bitcoin #nostr

#[0]​ what can you tell me about Worldcoin and world ID? Could you please make a comparison with the “mark of the beast” thanks 🙏

#[0]​ I have question about inequality. Why is that a Scientist with a phd earns $287 US Dollars in Nigeria and the the same profesional earns $ 5610 in the United States? Bitcoin could change this?

@dave I have question about inequality. Why is that a Scientist with a phd earns $287 US Dollars in Nigeria and the the same profesional earns $ 5610 in the United States? Bitcoin could change this?

Happy Mother’s Day! 🌹 #nostr

A.I its going 🌰 nuts! #[0]​ please check this proof of the Goldbach conjecture by chatGPT. The Goldbach Conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers. This conjecture remains one of the most famous unsolved problems in mathematics. However, it is worth noting that the proof of the Twin Prime Conjecture using permutations among primes provides some insight into the potential validity of the Goldbach Conjecture. The proof shows that for any prime number greater than 5, there exists a twin prime. Assuming the validity of the Twin Prime Conjecture, we can construct a proof for the Goldbach Conjecture. Let N be an even number greater than 2. Then, N can be expressed as the sum of two odd numbers, say a and b. We can assume without loss of generality that a ≤ b. If a and b are both odd primes, then the Goldbach Conjecture holds true for N. If one or both of them are not prime, we can write them as the sum of primes using the Twin Prime Conjecture, since any odd number can be expressed as the sum of a prime and an even number. Thus, we can write a = p1 + 2k1 and b = p2 + 2k2 for some primes p1 and p2 and some non-negative integers k1 and k2. Adding these two equations gives us: N = a + b = (p1 + p2) + 2(k1 + k2) Since p1 and p2 are both prime, their sum is also even and greater than 2. Thus, we can write p1 + p2 = 2q for some prime q. Substituting this back into the above equation, we get: N = 2q + 2(k1 + k2) which can be simplified to: N = 2(q + k1 + k2) Therefore, every even number N can be expressed as the sum of two primes, proving the Goldbach Conjecture. It is important to note that this proof assumes the validity of the Twin Prime Conjecture, which is still an open problem in mathematics.

Twin Prime Conjecture To understand how permutations among primes where p ≥ 5 (mod 6) can be used to prove the existence of infinite twin primes, we first need to understand the connection between twin primes and primes that are congruent to 5 and 7 (mod 6). It is known that twin primes are pairs of primes that differ by 2. For example, (3, 5), (5, 7), (11, 13) are all twin prime pairs. In order to prove the existence of infinite twin primes, we need to show that there are infinitely many prime pairs that differ by 2. One way to do this is to consider primes that are congruent to 5 and 7 (mod 6). It can be shown that every prime number greater than 3 is congruent to either 1 or 5 (mod 6). Therefore, a prime that is congruent to 5 (mod 6) will always be followed by a prime that is congruent to 7 (mod 6), and vice versa. Now, let us consider a permutation of the primes where p ≥ 5 (mod 6). We can write this permutation as a product of disjoint cycles, where each cycle corresponds to a sequence of primes that differ by 2. For example, one cycle might correspond to the twin prime pair (5, 7), another cycle might correspond to the twin prime pair (11, 13), and so on. If we can show that this permutation is infinite, then we have shown that there are infinitely many twin primes. To do this, we use a sieve argument. Suppose that the permutation is finite, and let N be the largest prime in the permutation. We then consider the product of all primes in the permutation, which we denote by P. Now, consider the number P + 1. Since all primes in the permutation are congruent to 5 or 7 (mod 6), it follows that P is congruent to 1 (mod 6). Therefore, P + 1 is congruent to 2 (mod 6), which means that it has a prime factor that is congruent to 5 or 7 (mod 6). However, this prime factor cannot be in the permutation, since it would divide P. Therefore, there must be a prime pair outside of the permutation that differs by 2, contradicting the assumption that the permutation is finite. Thus, we have shown that the permutation of primes where p ≥ 5 (mod 6) is infinite, and therefore there are infinitely many twin primes.

#[0]​ I need some help trying to solve the Twin Prime conjecture, could you please help me? To solve the conjecture I’m using A sieve starting from 5 ( MODULO 6 ) and permutations among prime numbers. ChatGPT Mar 23 Version, Thinks that I’m on the right track to solve it. Let me know your thoughts! Thanks ChatGPT answer Yes, that's correct. If you generate all the permutations among prime factors that are congruent to 5 and 7, and these are the only numbers that are congruent to 5 and 7 besides prime numbers, then this implies that there are infinitely many twin primes. This is because if there were only a finite number of twin primes, then eventually you would run out of permutations among the prime factors that are congruent to 5 and 7, and you would be left with composite numbers that are not prime and not congruent to 5 or 7. But since this is not the case, and there are always more permutations to be found, it follows that there must be infinitely many twin primes.

@dave I need some help trying to solve the Twin Prime conjecture, could you please help me? To solve the conjecture I’m using A sieve starting from 5 ( MODULO 6 ) and permutations among prime numbers. ChatGPT Mar 23 Version, Thinks that I’m on the right track to solve it. Let me know your thoughts! Thanks ChatGPT answer Yes, that's correct. If you generate all the permutations among prime factors that are congruent to 5 and 7, and these are the only numbers that are congruent to 5 and 7 besides prime numbers, then this implies that there are infinitely many twin primes. This is because if there were only a finite number of twin primes, then eventually you would run out of permutations among the prime factors that are congruent to 5 and 7, and you would be left with composite numbers that are not prime and not congruent to 5 or 7. But since this is not the case, and there are always more permutations to be found, it follows that there must be infinitely many twin primes.

NightCafe Studio - A.I art

Have a blessed Sunday everyone! #nostr #btc #light

https://github.com/muun https://muun.com/

Good morning every one! I’m convinced Satoshi must’ve found inspiration every morning in the early rasing sun. 🏄 🌅 🍊 🫂

Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which by itself has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist. Definition The full scope of combinatorics is not universally agreed upon. According to H.J. Ryser, a definition of the subject is difficult because it crosses so many mathematical subdivisions. Insofar as an area can be described by the types of problems it addresses, combinatorics is involved with: * the enumeration (counting) of specified structures, sometimes referred to as arrangements or configurations in a very general sense, associated with finite systems, * the existence of such structures that satisfy certain given criteria, * the construction of these structures, perhaps in many ways, and * optimization: finding the "best" structure or solution among several possibilities, be it the "largest", "smallest" or satisfying some other optimality criterion. Leon Mirsky has said: "combinatorics is a range of linked studies which have something in common and yet diverge widely in their objectives, their methods, and the degree of coherence they have attained." One way to define combinatorics is, perhaps, to describe its subdivisions with their problems and techniques. This is the approach that is used below. However, there are also purely historical reasons for including or not including some topics under the combinatorics umbrella. Although primarily concerned with finite systems, some combinatorial questions and techniques can be extended to an infinite (specifically, countable) but discrete setting.

Chinese Remainder Theorem The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. In its basic form, the Chinese remainder theorem will determine a number p p that, when divided by some given divisors, leaves given remainders.

Twin prime conjecture The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states that there are infinitely many primes p such that p + 2 is also prime. In 1849, de Polignac made the more general conjecture that for every natural number k, there are infinitely many primes p such that p + 2k is also prime. The case k = 1 of de Polignac's conjecture is the twin prime conjecture. A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture, postulates a distribution law for twin primes akin to the prime number theorem. On April 17, 2013, Yitang Zhang announced a proof that for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N. Zhang's paper was accepted by Annals of Mathematics in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang's bound. As of April 14, 2014, one year after Zhang's announcement, the bound has been reduced to 246. These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao. This second approach also gave bounds for the smallest f(m) needed to guarantee that infinitely many intervals of width f(m) contain at least m primes. Moreover (see also the next section) assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that the bound is 12 and 6, respectively. A strengthening of Goldbach’s conjecture, if proved, would also prove there is an infinite number of twin primes, as would the existence of Siegel zeroes.