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Prime Numbers and How to Avoid Them Pi Mu Epsilon April 19, 2012 CWRU Sequences including primes • 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, … • 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,… • 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, … Avoiding Primes • 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,… That almost works. If we start with a non-prime, we can avoid primes entirely: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, … But these numbers are all divisible by a common factor. That’s too easy. Arithmetic sequences • Suppose we look only at sequences of the form {An+B} , Where A and B have no common factor. • 199, 409, 619, 829,1039, 1249, 1459, 1669, 1879,2089, 2299, 2509, 2719, 2929, … This was discovered by Edward Escott in 1910. Green and Tao (2004) have proved that k consecutive primes occur in some arithmetic sequence, for any k. But we want ONLY primes • Then by the prime number theorem arithmetic progressions will not work. Let’s try exponentially growing lists. • Numbers one less than a power of 2: • 1,3,7,15,31,63,127,255,511,1023,2047,4095, 8191, 16383, 32767,65535, 131071,262143, 524287, 1048575,… Mersenne Add one instead Finally getting somewhere? Fermat Numbers • 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, … Nice try, Pierre! What went wrong? The order of 2 • So Euler saw he only had to test divide by 65, 129, 193, 257, …, 64k+1 In fact, he only needed to divide by primes in this list. When k=10 it works. Looking for Fermat Primes • By generalizing this argument, it can be seen that testing for Fermat primes requires looking at prime numbers of the form A New Question Sierpinski’s Theorem (1960) Proof of Sierpinski’s Theorem 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 The integers can be divided into seven sets: 2n, 4n+1, 8n+3, 16n+7, 32n+31, 64n+15, 64n+47 Proof of Sierpinski’s Theorem Now the plan is to divide the exponents into those seven groups, and for each group find a prime that always divides numbers whose exponents belong to that group. Proof of Sierpinski’s Theorem So when k is any number of the form 3i+2, half the terms in the sequence will be divisible by 3. Proof of Sierpinski’s Theorem Example: k=5 n=1,2,3,… 11, 21, 41, 81, 161, 321, 641, 1281, 2561, 5121,… So when k is any number of the form 5i+2, a quarter of the terms in the sequence will be divisible by 5. Proof of Sierpinski’s Theorem When k=17, these happen at the same time: 35, 69, 137, 273, 545, 1089, 2177, 4353, 8705,17409, 34817, 69633, 139265, 278529, 557057, 1114113, 2228225, 4456449,… Three quarters of the terms are divisible by 3 or 5, but occasionally a prime keeps showing up. Proof of Sierpinski’s Theorem Proof of Sierpinski’s Theorem Proof of Sierpinski’s Theorem Proof of Sierpinski’s Theorem Proof Concluded The “Chinese Remainder Theorem” guarantees we can find infinitely many such numbers. The smallest is k= 201 44650 31451 65117 This is a “Sierpinski number”, which makes every term in the infinite sequence composite. Half are divisible by 3, a quarter divisible by 5, etc. What is the smallest Sierpinski number? • In 1962, John Selfridge found the number 78577 (= 17 x 4621). It is conjectured that this is the smallest such number. • To rule out a smaller value of k it is necessary to find a prime number for some n and this k. • As of March 2002 there were only 17 smaller values for which no prime had been found. Seventeen or Bust • The Seventeen or Bust project was started in 2002 by two undergraduates. • As of October 2007, eleven of the possible counterexamples were ruled out by finding primes, the largest of which has almost four million digits. • Five of the eleven have been named Stephen Colbert primes. This Man Has Your Number Seventeen or Bust