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Myshkin
Myshkin
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Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$$$2. \quad(1764n-139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Carmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$) for the first expression. $C_i$ indicates the $i$-th Carmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

For the other expression,Edit. See the only prime triplet (again, $n<11$) is $5431\cdot 6983 \cdot 8534657$,comment of Zhiyun Cheng below for $n=3$ (if someone can confirm it to be Carmichael I'll edit)a translation of the text in chinese.

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Carmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th Carmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

For the other expression, the only prime triplet (again, $n<11$) is $5431\cdot 6983 \cdot 8534657$, for $n=3$ (if someone can confirm it to be Carmichael I'll edit).

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n-139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Carmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Added data for very small values ($n<11$) for the first expression. $C_i$ indicates the $i$-th Carmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

Edit. See the comment of Zhiyun Cheng below for a translation of the text in chinese.

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Myshkin
Myshkin
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Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a CharmichaelCarmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th CharmichaelCarmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

For the other expression, the only prime triplet (again, $n<11$) is $5431\cdot 6983 \cdot 8534657$, for $n=3$ (if someone can confirm it to be Carmichael I'll edit).

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Charmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th Charmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Carmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th Carmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

For the other expression, the only prime triplet (again, $n<11$) is $5431\cdot 6983 \cdot 8534657$, for $n=3$ (if someone can confirm it to be Carmichael I'll edit).

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Myshkin
Myshkin
  • 17.6k
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  • 137

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Charmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th Charmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Charmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Apparently it is an alternative proof of the infinitude of Carmichael numbers.

The other proof mentioned in the articles ("done by academics 20 years ago") is:

As for the method, unless someone took notes of the lecture at Zhejiang University, I guess it is unpublished at this point (except for the fragments shown in the news).

As a side note, notice two equations shown in one the photos:

enter image description here

$$1. \quad(6n+1)(18n+1)(54n^2+12n+1)$$ $$2. \quad(1764n+139)(2268n+179)(1000188n^2-157752n+6221)$$

This looks very similar to Chernick's result that $(6n+1)(12n+1)(18n+1)$ is a Charmichael number if each of the factors is prime. It is open whether this family of Charmichael numbers is infinite or not.

This is pure speculation, but the non-linear factors in Yu's numbers might make a big difference, since that is not a Dickinson's conjecture-type problem anymore.

Edit. Added data for very small values ($n<11$). $C_i$ indicates the $i$-th Charmichael number.

\begin{array}{|c|c|c|c|} \hline \mathrm{n}& \mathrm{eq. 1 } & \\ \hline 1 & 7 \cdot 19\cdot 67 &8911=C_7\\ \hline 2 & 13 \cdot 37\cdot 241 &115921=C_{18}\\ \hline 3 & - &-\\ \hline 4 & - &-\\ \hline 5 & - &-\\ \hline 6 & 37 \cdot 109\cdot 2017& 8134561=C_{93}\\ \hline 7 & 43 \cdot 127\cdot 2731& 14913991=C_{125}\\ \hline 8 & - & -\\ \hline 9 & - & -\\ \hline 10 & 61 \cdot 181\cdot 5521& 60957361=C_{209}\\ \hline \end{array}

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Myshkin
Myshkin
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Myshkin
Myshkin
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Myshkin
Myshkin
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