Impossible Exponential Diophantine Equation

Problem Statement

Explain why there is no integer $n$ such that $(n+1)^{41} - 79n^{40} = 2$ .

Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

Solutions

Proof by Modular Analysis and Case Checking

Step 1: Test $n = 0$

If $n = 0$ :

(0+1)^{41} - 79(0)^{40} = 1 - 0 = 1 \neq 2

So $n = 0$ is not a solution.

Step 2: Analyze for $n \neq 0$ using modular arithmetic

For $n \neq 0$ , consider the equation modulo $n$ :

By the Binomial Theorem:

(n+1)^{41} = \sum_{k=0}^{41} \binom{41}{k} n^k \cdot 1^{41-k}

All terms contain $n$ except the last term ( $k=0$ ):

(n+1)^{41} \equiv 1 \pmod{n}

The term $-79n^{40}$ is clearly divisible by $n$ :

-79n^{40} \equiv 0 \pmod{n}

Therefore:

\begin{aligned} (n+1)^{41} - 79n^{40} &\equiv 2 \pmod{n} \\ 1 - 0 &\equiv 2 \pmod{n} \\ 1 &\equiv 2 \pmod{n} \end{aligned}

Step 3: Interpret the congruence

$1 \equiv 2 \pmod{n}$ means $n \mid (1-2)$ , so $n \mid (-1)$ .

Therefore $n \in \{-1, 1\}$ .

Step 4: Test candidate values

For $n = 1$ :

(1+1)^{41} - 79(1)^{40} = 2^{41} - 79 = 2,199,023,255,552 - 79 \neq 2

For $n = -1$ :

(-1+1)^{41} - 79(-1)^{40} = 0 - 79(1) = -79 \neq 2

Conclusion

All possible integer values of $n$ have been checked and none satisfy the equation.

Therefore, there is no integer $n$ such that $(n+1)^{41} - 79n^{40} = 2$ .

\hfill $\square$

Takeaways

Modular Arithmetic Strategy: Working mod $n$ can dramatically constrain possible solutions
Binomial Expansion Mod $n$ : $(n+1)^k \equiv 1 \pmod{n}$ since all terms except constant contain $n$
Divisor Analysis: $1 \equiv 2 \pmod{n}$ means $n \mid (-1)$ , giving only $n \in \{-1, 1\}$
Exhaustive Checking: After constraining to finitely many cases, verify each directly

Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills:

HSC Proofs: https://vumaths.com/booklets/hsc-proofs/
HSC Vectors: https://vumaths.com/booklets/hsc-vectors/
HSC Last Resorts: https://vumaths.com/booklets/hsc-last-resorts/

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