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

1. P. Brown, Proof by Induction Test

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Copyright © 1996-2012 Alexander Bogomolny

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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Copyright © 1996-2012 Alexander Bogomolny