DULA

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.

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PRIME PERMUTATIONS P ≤ 5 ( MODULO 6) # # # # # # # # # # # # # # # # # # # import sympy from tkinter import * from tkinter import messagebox # Function to generate and display the prime factorizations of composite numbers def generate_factorizations(): # Get the custom limit from the input box limit = int(limit_input.get()) # Generate all the composite numbers congruent to 1, 5, and 7 modulo 6 up to the custom limit nums = [] for n in range(25, limit + 1): if n % 6 in (1, 5, 7, 11) and not sympy.isprime(n): nums.append(n) # Generate the prime factorization of each composite number factorizations = {} for n in nums: factorizations[n] = sympy.factorint(n) # Clear the existing output text and display the factorizations in the desired format output_text.delete('1.0', END) for n, factors in factorizations.items(): output_text.insert(END, f"{n} = {' x '.join(str(p) + '^' + str(e) for p, e in factors.items())}\n") # Check if all the factors are prime numbers all_primes = all(all(sympy.isprime(p) for p in factors.keys()) for factors in factorizations.values()) # Display a message box with the result if all_primes: messagebox.showinfo("All prime factors", "All factors in the output are prime numbers.") else: messagebox.showwarning("Not all prime factors", "Some factors in the output are not prime numbers.") # Create the main window and widgets root = Tk() root.title("Prime Factorization Generator P ≡ 1, 5, 7, 11 MOD 6") limit_label = Label(root, text="Enter a limit:") limit_label.grid(row=0, column=0) limit_input = Entry(root) limit_input.grid(row=0, column=1) generate_button = Button(root, text="Generate Factorizations", command=generate_factorizations) generate_button.grid(row=1, column=0, columnspan=2) output_label = Label(root, text="Prime factorizations:") output_label.grid(row=2, column=0) output_text = Text(root, height=20, width=50) output_text.grid(row=3, column=0, columnspan=2) root.mainloop()

P ≥ 5 (MOD6) & Permutations = Language of Universe.