The following is a series of apparently straightforward arithmetic problems for middle schoolers. However, the objective is to have students justify their reasoning beyond "guess and test" methods. Proving there is only one solution or none requires more careful logic using algebra as needed. Students will need some basic algebra for the "proofs." For the younger student, modify these questions to have them find the squares in questions 1,2,3 and 5. Take this as far as you wish...
In the following, square refers to the square of an integer. Justify your reasoning or prove each of the following.
(1) There is only one square which is 1 more than a prime.
(2) There is only one square which is 4 more than a prime.
(3) There is only one square which is 9 more than a prime.
(4) There is no square which is 16 more than a prime.
(5) There is only one square which is 25 more than a prime.
(6) Can one generalize this or not??
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Selected Answers, Solutions
(2) If n2 is 4 more than some prime, p, then we can write
p = n2 - 4 = (n-2)(n+2). Since p is prime, the smaller factor must be 1, so
n-2 = 1 or n = 3. Thus, there is only one square, 9, which is 4 more than the prime, 5.
(4) p = n2 - 16 = (n-4)(n+4). There n would have to equal 5, n2 would equal 25 but 25 - 16 = 9 is not prime.
(6) If there were a general rule would that mean we'd have a formula for primes?
Your thoughts about these questions...
Sunday, April 12, 2009