All Powers of 2 Are Equal to 1

We are going to prove by induction that,

For all integer n ≥ 0, 2 ⁿ = 1.

The claim is verified for n = 0; for indeed, 2⁰ = 1.

Assume the equation is correct for all n ≤ k, that is

2⁰ = 1, 2¹ = 1, 2² = 1, ..., 2^k = 1.

From these we now derive that also 2^k+1 = 1:

2^k+1 = 2^2k / 2^k-1 = 2^k × 2^k / 2^k-1 = 1×1/1 = 1.

Induction is complete.

What went wrong?

References

P. Brown, Proof by Induction Test

The error in the proof is subtle. The inductive step had to be formulated as

Assume the equation is correct for all 0 ≤ n ≤ k, where k ≥ 0.

The derivation then clearly fails for k = 0; for,

2¹ = 2⁰ / 2^-1 = 2¹,

which is 2^k+1 = 2^2k / 2^k-1 for k = 0. So 2¹ = 2¹ does not actually establish that 2¹ = 1. And this then can't be used in deriving 2² = 1, and so on.

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