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GIMPS has discovered a new largest known prime number: 2^{82589933}1 (24,862,048 digits) Just showing those entries submitted by 'Rupinski': (Click here to show all)
The greatest number of consecutive integers which are pairwise relatively prime. [Rupinski] The function n^(1/n) achieves its maximum value for integers n at n = 3. [Rupinski] According to the quadratic reciprocity theorem, if p and q are distinct primes of the form 4k+3, then exactly one of the following congruences is solvable in integers: x^2 = p mod q or x^2 = q mod p. [Rupinski] The only Sophie Germain Prime p of the form 4k+3 such that M_{p} is prime. All larger M_{p} for such p are composite as shown by Euler and Lagrange. [Rupinski] The smallest possible Proth prime, k*2^{n}+1 with k < 2^{n}, here k=1, n=1. [Rupinski]
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