Finite Evidence and a False Universal Claim
During a review of a student's independent project, a teacher finds the following written argument: 'Let f(n)=n^2+n+17. I evaluated f(0) through f(15) and obtained 17, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, and 257. Every one is prime. Therefore f(n) is prime for every nonnegative integer n.' The student has not shown the arithmetic for the largest values, and a classmate objects that 'a pattern is not a proof.' The teacher wants to use the episode to teach the difference between inductive evidence and deductive proof without discouraging the student's curiosity. The class has studied polynomial evaluation, prime and composite numbers, and conditional statements. The teacher also wants a generalization that is actually true and can be proved in one line.
1. Evaluate the student's conclusion. Identify the smallest nonnegative integer counterexample, show the arithmetic, and explain why the sixteen verified cases cannot establish the universal claim.
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The claim is false. Compute f(16)=16^2+16+17=256+16+17=289=17^2, which is composite. Since 16 is a nonnegative integer, one counterexample disproves the universal statement 'f(n) is prime for every n≥0.' The sixteen checks are evidence, not proof: a universal claim ranges over infinitely many cases, and no finite list can exhaust them. In logical form, the student has shown P(0) and P(1) and ... and P(15), which does not imply that P(n) holds for every n. The correct conclusion from the checks is only the bounded statement 'f(n) is prime for 0≤n≤15.'
2. Generalize the counterexample. For f(n)=n^2+n+c, show algebraically that f(c−1)=c^2 and f(c)=c(c+2). Use these identities to give a second composite value of the student's f and to explain why checking consecutive values can miss a composite.
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Expand f(c−1)=(c−1)^2+(c−1)+c = c^2−2c+1+c−1+c = c^2. Expand f(c)=c^2+c+c = c^2+2c = c(c+2). For c=17, f(16)=17^2=289 and f(17)=17·19=323, both composite. The identity shows composites are forced at n=c−1 and n=c for any c>1. Checking only 0 through 15 stops one step before the predictable composite at 16; a finite run can stop just before a structural break. The algebra explains why the pattern was not accidental.
3. Write a short response to the student that credits the pattern, states the logical status of the claim, and gives a valid next step. Include the exact language you would use and explain why that language is appropriate.
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Possible response: 'Your sixteen evaluations are a strong pattern and worth sharing. They show f(n) is prime for 0≤n≤15, but they do not prove the universal claim. In fact, f(16)=289=17^2, so the universal claim is false. A good next step is to prove a statement you can establish, such as: for f(n)=n^2+n+c, f(c−1)=c^2, so f(16) had to be composite. This turns the surprise into a proof.' This language credits the evidence, distinguishes a bounded statement from a universal one, supplies the counterexample, and redirects the student to a provable generalization. It avoids saying 'you were wrong' without honoring the reasoning.