Weyl's inequality

In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period. If the period of the decimal representation of a/p is 2n, so that

${\frac {a}{p}}=0.{\overline {a_{1}a_{2}a_{3}\dots a_{n}a_{n+1}\dots a_{2n}}}$

then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words

$a_{i}+a_{i+n}=9\,$

$a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.\,$

For example

${\frac {1}{17}}=0.{\overline {0588235294117647}}{\text{ and }}05882352+94117647=99999999.\,$

Midy's theorem in other bases

Midy's theorem and its extension do not depend on special properties of the decimal expansion, but work equally well in any base b, provided we replace 10^k − 1 with b^k − 1 and carry out addition in base b. For example, in octal

${\frac {1}{19}}=0.{\overline {032745}}_{8}$

$032_{8}+745_{8}=777_{8}\,$

$03_{8}+27_{8}+45_{8}=77_{8}.\,$

Proof of Midy's theorem

Short proofs of Midy's theorem can be given using results from group theory. However, it is also possible to prove Midy's theorem using elementary algebra and modular arithmetic:

Let p be a prime and a/p be a fraction between 0 and 1. Suppose the expansion of a/p in base b has a period of ℓ, so

${\begin{aligned}&{\frac {a}{p}}=[0.{\overline {a_{1}a_{2}\dots a_{\ell }}}]_{b}\\[6pt]&\Rightarrow {\frac {a}{p}}b^{\ell }=[a_{1}a_{2}\dots a_{\ell }.{\overline {a_{1}a_{2}\dots a_{\ell }}}]_{b}\\[6pt]&\Rightarrow {\frac {a}{p}}b^{\ell }=N+[0.{\overline {a_{1}a_{2}\dots a_{\ell }}}]_{b}=N+{\frac {a}{p}}\\[6pt]&\Rightarrow {\frac {a}{p}}={\frac {N}{b^{\ell }-1}}\end{aligned}}$

where N is the integer whose expansion in base b is the string a₁a₂...a_ℓ.

Note that b ^ℓ − 1 is a multiple of p because (b ^ℓ − 1)a/p is an integer. Also bⁿ−1 is not a multiple of p for any value of n less than ℓ, because otherwise the repeating period of a/p in base b would be less than ℓ.

Now suppose that ℓ = hk. Then b ^ℓ − 1 is a multiple of b^k − 1. (To see this, substitute x for b^k; then b^ℓ = x^h and x − 1 is a factor of x^h − 1. ) Say b ^ℓ − 1 = m(b^k − 1), so

${\frac {a}{p}}={\frac {N}{m(b^{k}-1)}}.$

But b ^ℓ − 1 is a multiple of p; b^k − 1 is not a multiple of p (because k is less than ℓ ); and p is a prime; so m must be a multiple of p and

${\frac {am}{p}}={\frac {N}{b^{k}-1}}$

is an integer. In other words

$N\equiv 0{\pmod {b^{k}-1}}.\,$

Now split the string a₁a₂...a_ℓ into h equal parts of length k, and let these represent the integers N₀...N_h − 1 in base b, so that

${\begin{aligned}N_{h-1}&=[a_{1}\dots a_{k}]_{b}\\N_{h-2}&=[a_{k+1}\dots a_{2k}]_{b}\\&{}\ \ \vdots \\N_{0}&=[a_{l-k+1}\dots a_{l}]_{b}\end{aligned}}$

To prove Midy's extended theorem in base b we must show that the sum of the h integers N_i is a multiple of b^k − 1.

Since b^k is congruent to 1 modulo b^k − 1, any power of b^k will also be congruent to 1 modulo b^k − 1. So

$N=\sum _{i=0}^{h-1}N_{i}b^{ik}=\sum _{i=0}^{h-1}N_{i}(b^{k})^{i}$

$\Rightarrow N\equiv \sum _{i=0}^{h-1}N_{i}{\pmod {b^{k}-1}}$

$\Rightarrow \sum _{i=0}^{h-1}N_{i}\equiv 0{\pmod {b^{k}-1}}$

which proves Midy's extended theorem in base b.

To prove the original Midy's theorem, take the special case where h = 2. Note that N₀ and N₁ are both represented by strings of k digits in base b so both satisfy

$0\leq N_{i}\leq b^{k}-1.\,$

N₀ and N₁ cannot both equal 0 (otherwise a/p = 0) and cannot both equal b^k − 1 (otherwise a/p = 1), so

$0<N_{0}+N_{1}<2(b^{k}-1)\,$

and since N₀ + N₁ is a multiple of b^k − 1, it follows that

$N_{0}+N_{1}=b^{k}-1.\,$

References

• Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158-160, 1957.
• E. Midy, De Quelques Propriétés des Nombres et des Fractions Décimales Périodiques.

College of Nantes, France: 1836.

• Ross, Kenneth A. Repeating decimals: a period piece. Math. Mag. 83 (2010), no. 1, 33–45.

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