Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(-q;\,-q)_\infty}{(q;\,q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical (Simon Plouffe, Feb. 20, 2011): $$\sum_{n=1}^{\infty}e^{-\pi(n-1)}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(-q;\,-q)_\infty}{(q;\,q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical (Simon Plouffe, Feb. 20, 2011): $$\sum_{n=0}^{\infty}e^{-\pi n}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(q;\,q)_\infty}{(-q;\,-q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical (Simon Plouffe, Feb. 20, 2011): $$\sum_{n=1}^{\infty}e^{-\pi(n-1)}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(-q;\,-q)_\infty}{(q;\,q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical (Simon Plouffe, Feb. 20, 2011): $$\sum_{n=1}^{\infty}e^{-\pi(n-1)}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(q;\,q)_\infty}{(-q;\,-q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical : sum(exp(-Pi)^(n-1)*a(n),n=1..infinity) = 2^(1/8). Simon Plouffe, Feb. 20, 2011.

I did some numerical experiments related to this observation, and their results suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?

Recall that $(a;\,q)_\infty$ is the $q$-Pochhammer symbol: $$(a;\,q)_\infty=\prod_{n=0}^\infty(1-a \, q^n).\tag1$$ Its important special case $(q;\,q)_\infty=\prod_{n=1}^\infty(1-q^n)$ is sometimes called the Euler function. It appears in Euler's pentagonal number theorem, and its reciprocal $(q;\,q)_\infty^{-1}$ is the generating function for the partition numbers. It is also related to Jacobi theta functions and Ramanujan theta functions.

Let $$f(x) = \frac{(q;\,q)_\infty}{(-q;\,-q)_\infty},\quad\text{where}\,\,q=e^{-\pi\sqrt x}.\tag2$$ In the OEIS entry A080054 there is an empirical observation by Simon Plouffe that apparently $f(1)=\sqrt[8]2$.

Empirical (Simon Plouffe, Feb. 20, 2011): $$\sum_{n=1}^{\infty}e^{-\pi(n-1)}a(n) =\sqrt[8]2.$$

I did some numerical experiments related to this observation, and the outcomes suggest a fascinating stronger conjecture:

Conjecture: For every $p\in\mathbb Q,\,p>0$, the value $f(p)$ is an algebraic number.

For example, it appears that $$f(3/5) = \sqrt[8]{2} \cdot \sqrt[4]{9 \sqrt{5}+5 \sqrt{15}-11 \sqrt{3}-19},$$ and $f(13/7)$ is an algebraic number of degree $96$ whose minimal polynomial is $$x^{96}-647442063456 \, x^{88}+16702438371168 \, x^{80}-529345497357824 \, x^{72}+4159684203040512 \, x^{64}-12099397290541056 \, x^{56}+16408771708010496 \, x^{48}-10607690933600256 \, x^{40}+2651923007078400 \, x^{32}-367001600 \, x^{24}+257949696 \, x^{16}-100663296 \, x^8+16777216$$ and an isolating rational interval is $(37/36,\,6/5)$.

Is this conjecture new? Is it known to be true? If not, can you suggest any ideas how to (dis-)prove it?