Fermat's Last Theorem: the same question one power up

What to assume they know

• Baudhāyana-Pythagoras triples, and how to make infinitely many — integer triples, and the two arguments that there are infinitely many
• Powers with whole-number exponents: what x³ and x⁴ mean, and how to compute small ones
• Natural numbers, and that a solution here means all three of x, y, z being natural
• Substituting into an equation to test a candidate, and the difference between a case that works and a statement that always holds
• That a general statement about all whole numbers cannot be settled by checking cases, however many — the habit §2.5 established when it substituted k

What they should be able to do

• State the equation xⁿ + yⁿ = zⁿ and the conditions the chapter attaches to it: natural numbers, and an exponent above 2
• Explain how the question arose from the study of integer triples
• Contrast the case n = 2, with infinitely many solutions, against every case above it, with none
• Explain why checking cubes one after another can never settle the question, however many are checked
• Recount the chapter's account of Fermat's marginal note and what became of the proof it claimed
• Give the chapter's timeline: Fermat in the 17th century, over three centuries of failure, Wiles reading about the problem in 1963, and his proof in 1994
• Distinguish "no proof was found" from "no proof existed", and say which the chapter actually asserts
• Say why this section sits at the end of a chapter about Baudhāyana

Where it usually goes wrong

• "It was called a theorem, so Fermat proved it." The chapter is careful: the claimed proof was never found, and it was proved by someone else 300-odd years later. The name is historical. An explanation must not let "theorem" imply Fermat had one, and should not assert that he did not either — the chapter takes no position and neither should the explanation.
• "Nobody could find a solution, so there isn't one." Backwards, and it is the central logical point of the section. Not finding is not the same as not existing. This is why the near-miss 728 and 729 belongs in the explanation: it shows how close a search can come while proving nothing.
• "So mathematicians spent 300 years checking numbers." They did not, or at least that is not what took the time. Checking cases could never have finished the job. What was needed was an argument covering every exponent at once, which is why the problem was hard and why the eventual proof uses mathematics far beyond this chapter.
• "A ten-year-old proved Fermat's Last Theorem." He was ten when he read about it and an adult mathematician when he proved it, three decades later. The chapter says exactly this, and an explanation that compresses it turns a good story about persistence into a false one about precocity.
• "The theorem says powers never add up." It says nothing of the kind. Powers add up constantly; what fails is the demand that the total be the same power of a whole number. And for n = 1 and n = 2 even that succeeds endlessly. Section 10 exists to stop this.
• "There might still be a huge counterexample." Not any more; the statement is proved. Before 1994 that was a live possibility and it is part of why the problem resisted — the numbers involved could have been enormous.
• "This is examinable content." Treat it as context and culture rather than as technique. There is nothing here to compute. Its value is that it shows a student what an open problem is, and that the arithmetic they have just done sits at the edge of something unresolved for centuries.

Questions to check understanding

• Write down the equation and state the two conditions the chapter attaches to it
• Say how many natural-number solutions the equation has when the exponent is 2, and give one
• Explain, in two or three sentences, why testing cubes cannot settle the question
• Give the two dates the chapter attaches to Andrew Wiles, and say what happened at each
• Explain why the theorem's name is misleading about who proved it
• Given a claim about all whole numbers, say what would be needed to disprove it and what would be needed to prove it
• Short answer: what makes the exponent 2 special in this chapter?

Examples worth working on the board

This section is largely narrative, so the mathematical content has to be supplied by the explanation from the chapter's own material — everything below marked derived is added here.

• The starting point the chapter names (Part II, §2.6, p.50 and p.51). The study of Baudhāyana triples led Fermat, a French mathematician of the 17th century, to a general claim about sums of powers. The chapter states that infinitely many squares are the total of two squares — which is exactly what §2.5 established — and that this set Fermat wondering about cubes, fourth powers and beyond.
• The equation as printed (Part II, §2.6, p.51): xⁿ + yⁿ = zⁿ, with x, y and z natural numbers and n greater than 2. The chapter's SUMMARY on Part II p.54 writes the same statement with a, b and c.
• **The n = 2 supply, for contrast** (from §2.5, Part II pp.48–50). Any triple will do as the opening exhibit; (3, 4, 5) and (5, 12, 13) are the chapter's own, and the odd-square machine turns out more on demand. This is the abundance the section is about to take away.
• A cube search to run (derived, and the explanation needs it — the chapter supplies no numerical trial at all). The cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Add them in pairs and look for another cube on the list: 1 + 8 = 9; 8 + 27 = 35; 27 + 64 = 91; 64 + 125 = 189; 1 + 27 = 28; 1 + 64 = 65; 8 + 64 = 72; 125 + 216 = 341; 216 + 512 = 728, which misses 729 by one and is the single most useful near-miss to show; 512 + 729 = 1241. Nothing lands.
• The near-miss as the whole pedagogical point (derived). 728 sits one below 729. A student who sees that will understand immediately why searching cannot settle the matter: the next pair might be the one, and there is always a next pair.
• The margin note (Part II, §2.6, p.51). The chapter reports that Fermat wrote, in the margin of a book about the properties and patterns of whole numbers, that no such cube or fourth power or higher exists, and added a remark that he had a proof and that the margin was too small to hold it. Do not show the remark as a quotation; report it. The chapter then says the proof was never found.
• The artwork (Part II, §2.6, p.51, two panels side by side). On the left, a seventeenth-century man in a long wig at a desk, an open printed book in front of him and an inkwell beside it, a curtained window behind. On the right, a fair-haired boy in a school sweater at a desk with an open exercise book, a pencil in his hand and a closed book beside him, a window behind. The pairing is the argument of the section in one image: the same problem, three hundred years apart. Both drawn books carry simulated writing — ruled marks on the open pages, and two lines of title lettering with a small cover picture on the closed book beside the boy — and none of it resolves into words even on the panels. So there is nothing in the artwork to transcribe, and a redrawn version should likewise indicate text rather than spell any out.
• The timeline the chapter gives (Part II, §2.6, p.51). Fermat lived in the 17th century. After his death many mathematicians attempted a proof; the chapter says more than three hundred years of attempts failed. In 1963 a ten-year-old, Andrew Wiles, read Eric Bell's The Last Problem about the theorem and its history, and resolved to prove it. He proved it in 1994.
• The boundary, for section 10 (derived, and it is the strongest available piece of mathematics in this topic). For n = 1 the equation is x + y = z, which has endlessly many natural solutions — 2 + 3 = 5, and any other pair you like. For n = 2 there are infinitely many, and §2.5 built two machines for them. For every n above 2 there are none. So the supply of solutions goes from unlimited, to unlimited, to nothing, and it stops exactly where this chapter's own subject stops. The chapter never lays the three cases side by side; doing so is the explanation's best contribution.
• The SUMMARY line (Part II, p.54). It states the theorem, names it, and credits Wiles with the 1994 proof.

Figures to have open

• A three-column comparison for section 10: exponent 1, exponent 2, exponent above 2, with the count of natural-number solutions under each. Not printed in the chapter, and it is the figure that turns this topic from anecdote into mathematics.
• A cube-addition table for section 3: the first ten cubes, a column of pair totals, and the cube list alongside for comparison, so the near-miss at 728 is visible. Not printed in the chapter; the chapter runs no numerical trial.
• A timeline for sections 6 to 9, carrying only what the chapter states: the 17th century, more than three hundred years of attempts, 1963, 1994. Do not add dates the chapter does not give.
• The two-panel illustration pairing Fermat at his book with the boy at his exercise book (Part II, §2.6, p.51). Redraw as original artwork; do not reproduce the printed illustration. It carries simulated lettering — ruled marks on the open pages and a two-line title with a small cover picture on the closed book beside the boy — none of which resolves into words. Indicate text; do not spell any out.
• No photograph is needed, and no portrait of a real person should be invented — the chapter's own images are drawings and the explanation's should be too.

Where this sits in the book

• NCERT Ganita Prakash Class 8, Part II, printed Chapter 2, "The Baudhāyana-Pythagoras Theorem", §2.6 "A Long-Standing Open Problem", Part II pp.50–51. The section opens at the foot of Part II p.50 and runs to the end of Part II p.51.
• Backward dependency inside the chapter: §2.5 at Part II pp.48–50 supplies the abundance of square solutions that the section reacts against.
• The chapter's SUMMARY at Part II p.54 states the theorem with the letters a, b, c and credits the 1994 proof.
• Books the chapter names: Euclid's Elements at Part II p.39, in the earlier material on √2; Eric Bell's The Last Problem at Part II p.51.