By J J Sakurai
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Extra info for Invariance Principles and Elementary Particles (Invest. in Physics)
Thus the last digit is 3. 184 Example Prove that every year, including any leap year, has at least one Friday 13th. Solution: It is enough to prove that each year has a Sunday the 1st. ) Now, each remainder class modulo 7 is represented in the third column, thus each year, whether leap or not, has at least one Sunday the 1st. 185 Example Find infinitely many integers n such that 2n + 27 is divisible by 7. Solution: Observe that 21 ≡ 2, 22 ≡ 4, 23 ≡ 1, 24 ≡ 2, 25 ≡ 4, 26 ≡ 1 mod 7 and so 23k ≡ 1 mod 3 for all positive integers k.
Solution: There are no such solutions. All perfect fourth powers mod 16 are ≡ 0 or 1 mod 16. This means that n41 + · · · + n414 can be at most 14 mod 16. But 1599 ≡ 15 mod 16. 190 Example (P UTNAM 1986) What is the units digit of 1020000 ? 10100 + 3 52 Chapter 3 Solution: Set a − 3 = 10100. Then [(1020000)/10100 + 3] = [(a − 3)200/a] = 1 200 200 200−k 200 k 200 199−k = (−3)k. Since 200 [ a (−3)k] = 199 k=0(−1) k=0 k a k a k=0 k 200 199−k k 200 (−3)k ≡ = −3199. As a ≡ 3 mod 10, 199 0, (3)199 199 k=0 k a k=0(−1) k k 200 ≡ −3199 ≡ 3 mod 10.
Each factor is greater than 1 for n > 1, and so n4 + 4 cannot be a prime. 39 Some Algebraic Identities 133 Example Find all integers n ≥ 1 for which n4 + 4n is a prime. Solution: The expression is only prime for n = 1. Clearly one must take n odd. For n ≥ 3 odd all the numbers below are integers: n4 + 22n = n4 + 2n22n + 22n − 2n22n 2 = (n2 + 2n)2 − n2(n+1)/2 = (n2 + 2n + n2(n+1)/2)(n2 + 2n − n2(n+1)/2). It is easy to see that if n ≥ 3, each factor is greater than 1, so this number cannot be a prime.