Three paths converge

When does the equation

x² + 7 = 2ⁿ

have integer solutions?

This is an interesting question, but why would anyone ask it? This post looks at three paths that have led to this problem.

Ramanujan

Ramanujan [1] considered this problem in 1913. He found five solutions and conjectured that there were no more. Then in 1959 three authors [2] wrote a paper settling the conjecture, showing that Ramanujan was right. We don't know what motivated Ramanujan, but the subsequent paper was a response to Ramanujan.

Nagell

T. Nagell [3] published a solution in 1960 after becoming aware of [2]. This paper republished in English a solution the author had first published in 1948 in a Norwegian journal. Nagell says he gave the problem as an exercise in a number theory textbook he wrote in 1951. By mentioning his textbook but not Ramanujan, Nagell implies that he came to the problem independently.

Golay

I ran into the problem a week ago in the course of looking at a problem that came out of Golay's work in coding theory. A necessary condition for the existence of a perfect binary code of length n including p redundant bits that can detect up to 2 errors is

This leads, via the quadratic equation, to the equation at the top of the post.

All solutions

Each of the paths above states the problem in different notation; it's simpler to state the solutions without variables.

• 1² + 7 = 2³
• 3² + 7 = 2⁴
• 5² + 7 = 2⁵
• 11² + 7 = 2⁷
• 181² + 7 = 2¹⁵

Verifying that these are solutions is trivial. Proving that there are no more solutions is not trivial.

References

[1] K. J. Sanjana and T. P. Trivedi, J. Indian Math. Soc. vol. 5 (1913) pp. 227-228.

[2] Th. Skolem, S. Chowla and D. J. Lewis. The Diophantine Equation 2ⁿ⁺² - 7 = x². Proceedings of the American Mathematical Society , Oct., 1959, Vol. 10, No. 5. pp. 663-669

[3] T. Nagell, The Diophantine Equation x² + 7 = 2ⁿ. Arkiv for Matematik, Band 4 nr 13. Jan 1960.

Thanks to Brian Hopkins for his help on this problem via his answer to my question on Math Overflow.

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