• Which of ²²²Rn and ²²²Fr has lower energy?
• Which of the ${\displaystyle 4^{-}}$ state and the ${\displaystyle 9^{-}}$ state of ²¹⁴At has lower energy? (levels of ²¹⁴At)
• Which of ²⁴⁷Cm and the ${\displaystyle 7/2^{+}}$ state of ²⁴⁷Bk has lower energy? (levels of ²⁴⁷Bk)
• Which of ²⁵⁹Fm and ²⁵⁹Md has lower energy?

Note the error margins! These questions do not have decisive answers by now.

Important: Please kindly contact me by jianing.song@polytechnique.edu. My old email address sjn1508@163.com is no longer in use.

Table of minimal polynomial of -2 cos(2π/n) (note the minus sign!)

PARI code giving the minimal polynomial of ${\displaystyle -2\cos {\frac {2\pi }{n}}}$:

P(n) = prod(m=1, n/2, (x + 2*cos(2*m*Pi/n))^(gcd(n,m)==1))

This is not most satisfactory because it doesn't give an integral polynomial, but it gives the correct result better than the apparently "obvious" solution

P(n) = algdep(-2*cos(2*Pi/n), eulerphi(n)/2)

which works for n ≤ 25, but then breaks down because a false polynomial nullifies ${\displaystyle -2\cos {\frac {2\pi }{n}}}$ with smaller floating error.

n	Pn	Pn(k) for k = 3	k = 4	k = 6
3	${\displaystyle X-1}$	2	3	5
4	${\displaystyle X}$	3	4	6
5	${\displaystyle X^{2}-X-1}$	5	11	29
6	${\displaystyle X+1}$	4	5	7
7	${\displaystyle X^{3}-X^{2}-2X+1}$	13	41	169
8	${\displaystyle X^{2}-2}$	7	14	34
9	${\displaystyle X^{3}-3X-1}$	17	51	197
10	${\displaystyle X^{2}+X-1}$	11	19	41
11	${\displaystyle X^{5}-X^{4}-4X^{3}+3X^{2}+3X-1}$	89	571	5741
12	${\displaystyle X^{2}-3}$	6	13	33
13	${\displaystyle X^{6}-X^{5}-5X^{4}+4X^{3}+6X^{2}-3X-1}$	233	2131	33461
14	${\displaystyle X^{3}+X^{2}-2X-1}$	29	71	239
15	${\displaystyle X^{4}+X^{3}-4X^{2}-4X+1}$	61	241	1345
16	${\displaystyle X^{4}-4X^{2}+2}$	47	194	1154
17	${\displaystyle X^{8}-X^{7}-7X^{6}+6X^{5}+15X^{4}-10X^{3}-10X^{2}+4X+1}$	1597	29681	1136689
18	${\displaystyle X^{3}-3X+1}$	19	53	199
19	${\displaystyle X^{9}-X^{8}-8X^{7}+7X^{6}+21X^{5}-15X^{4}-20X^{3}+10X^{2}+5X-1}$	4181	110771	6625109
20	${\displaystyle X^{4}-5X^{2}+5}$	41	181	1121

Let p be an odd prime not dividing k² - 4, then there exists a unique number of the form ${\displaystyle n={\dfrac {p-\left({\dfrac {k^{2}-4}{p}}\right)}{m}}\geq 3}$ such that p divides P_n(k). The parity of m is determined by the value of ${\displaystyle \left({\dfrac {-(k-2)}{p}}\right)}$. Consider the case k = 4, so k² - 4 = 12 and -(k - 2) = -2:

p	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
corresponding m	1	1	2	1	2	2	1	1	1	1	6	2	1	3	2	1	4	5	2	1	2	2	6
${\displaystyle \left({\dfrac {-(k-2)}{p}}\right)}$	-1	-1	1	-1	1	1	-1	-1	-1	-1	1	1	-1	-1	1	-1	1	-1	1	-1	1	1	1

Replacing ${\displaystyle \mathbb {R} }$ and ${\displaystyle \mathbb {C} }$ in linear algebra

Traditionally, various topics on linear algebra assume that the base field of study is either ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$. But wait, we put too many restrictions on ourselves!

Definition. An ordered field is called archimedean if the integers are unbounded. An ordered field is called an Euclidean field if every positive element is a square. A field is called quadratically closed if every element is a square. A field ${\displaystyle K}$ is called real-closed if ${\displaystyle -1}$ is not a square ${\displaystyle K}$, and ${\displaystyle K({\sqrt {-1}})}$ is algebraically closed.

We note that a field ${\displaystyle K}$ can be turned into an Euclidean field if and only if ${\displaystyle -1}$ is not a square in ${\displaystyle K}$, and ${\displaystyle K({\sqrt {-1}})}$ is quadratically closed.

Theory on inner product: ${\displaystyle \mathbb {R} }$ can be replaced by any archimedean Euclidean field. Technically inner products can be defined on every ordered field, but in order to allow Gram–Schmidt process we need that every positive element is a square, so it's better if the base field is Euclidean. Moreover, we want that a norm satisfying parallelogram law is induced by an inner product, so we want ${\displaystyle \mathbb {Q} }$ to be dense in the base field, which means that it's best if the base field is also archimedean. (If we do not care about norms, of course an Euclidean field is enough.) I didn't figure out how we could replace ${\displaystyle \mathbb {C} }$. I presume that it can be replaced by an arbitrary quadratically closed field with characteristic 0 and cardinality ${\displaystyle \leq {\mathfrak {c}}}$, but must such a field have an archimedean Euclidean subfield of index 2? (Anyway, must a quadratically closed field with characteristic 0 have an Euclidean subfield of index 2? Of course this is true for an algebraically closed field with characteristic 0: see the next paragraph.)

Theory on orthogonal/unitary diagonalization: ${\displaystyle \mathbb {R} }$ can be replaced by any real-closed field, on which every symmetric matrix is diagonalizable. In fact real-closedness is a sufficient but not necessary condition (see here), but for the sake of simplicity we will assume it. ${\displaystyle \mathbb {C} }$ can be replaced by any algebraically closed field with characteristic 0, because such a field must have a subfield of index 2 (which must be real closed; see here), so an automorphism of order 2 which looks like the ordinary complex conjugation.

Theory on binary forms: ${\displaystyle \mathbb {R} }$ can be replaced by any Euclidean field, which is precisely one of the two kinds of fields on which the Sylvester law of inertia holds (see here). This means that every symmetric matrix on an Euclidean field is congruent to a diagonal matrix consisting of 1s, (-1)'s, and 0s, and the occurence of each number is independent of the congruence factor. ${\displaystyle \mathbb {C} }$ can be replaced by any quadratically closed field (the other kind of fields on which the Sylvester law of inertia is valid; see the link aformentioned in this paragraph): every symmetric matrix on such a field is is congruent to a diagonal matrix consisting of 1s and 0s, where the number of 1s is equal to the rank of the matrix.

Character values of representations of finite group

Let ${\displaystyle K}$ be a field. What is the set of all numbers of the form ${\displaystyle \Theta _{\rho }(g)=\operatorname {Tr} \rho (g)}$, where ${\displaystyle \rho }$ is a representation of a finite group ${\displaystyle G}$ in ${\displaystyle K}$, and ${\displaystyle g}$ is an element in ${\displaystyle G}$?

Lemma. Let ${\displaystyle L}$ be an algebraically closed field, and ${\displaystyle S}$ be the set of elements in ${\displaystyle L}$ that can be expressed as a sum of roots of unity. If ${\displaystyle L}$ is of characteristic ${\displaystyle p}$, then ${\displaystyle S={\overline {\mathbb {F} }}_{p}}$, where ${\displaystyle \mathbb {F} _{p}}$ is the prime field of ${\displaystyle L}$ and ${\displaystyle {\overline {\mathbb {F} }}_{p}}$ is the algebraic closure of ${\displaystyle \mathbb {F} _{p}}$ within ${\displaystyle L}$. If ${\displaystyle L}$ is of characteristic 0, then ${\displaystyle S=\mathbb {Q} ^{\mathrm {ab} }\cap {\overline {\mathbb {Z} }}}$, where ${\displaystyle \mathbb {Q} }$ is the prime field of ${\displaystyle L}$, ${\displaystyle \mathbb {Q} ^{\mathrm {ab} }}$ is the extension of ${\displaystyle \mathbb {Q} }$ generated by all roots of unity, and ${\displaystyle {\overline {\mathbb {Z} }}}$ is the set of algebraic integers in ${\displaystyle L}$,

Proof. The inclusion ${\displaystyle \subset }$ is clear. For the case characteristic ${\displaystyle p}$, every element in ${\displaystyle {\overline {\mathbb {F} }}_{p}}$ is automatically a root of unity, so lies in ${\displaystyle S}$. For characteristic 0, each element in ${\displaystyle \mathbb {Q} ^{\mathrm {ab} }\cap {\overline {\mathbb {Z} }}}$ lies in some ${\displaystyle \mathbb {Q} (\zeta _{n})\cap {\overline {\mathbb {Z} }}=\mathbb {Z} [\zeta _{n}]}$, so it is a sum of roots of unity (possibly with repetitions).

Since ${\displaystyle g}$ is of finite order, ${\displaystyle \rho (g)}$ is also of finite order, so the eigenvalues of ${\displaystyle \rho (g)}$ in ${\displaystyle {\overline {K}}}$ (an algebraic closure of ${\displaystyle K}$) must be roots of unity, which means that a character value must be an element in ${\displaystyle K}$ which can be expressed as a sum of roots of unity in ${\displaystyle {\overline {K}}}$.

If ${\displaystyle K}$ is of characteristic ${\displaystyle p}$, let ${\displaystyle \mathbb {F} _{p}}$ be the prime field of ${\displaystyle K}$, then the lemma above tells us that a character value lies in ${\displaystyle K\cap {\overline {\mathbb {F} }}_{p}}$, which is the maximal algebraic subextension of ${\displaystyle \mathbb {F} _{p}}$ within ${\displaystyle K}$. Since every element in ${\displaystyle {\overline {\mathbb {F} }}_{p}}$ is itself a root of unity, every element in ${\displaystyle K\cap {\overline {\mathbb {F} }}_{p}}$ is automatically a character value of a representation of dimension 1 of a finite cyclic group in ${\displaystyle K}$.

If ${\displaystyle K}$ is of characteristic 0, let ${\displaystyle \mathbb {Q} }$ be the prime field of ${\displaystyle K}$, then the lemma above tells us that a character value lies in ${\displaystyle K\cap \mathbb {Q} ^{\mathrm {ab} }\cap {\overline {\mathbb {Z} }}}$ (note that ${\displaystyle K\cap \mathbb {Q} ^{\mathrm {ab} }}$ is the maximal abelian subextension of ${\displaystyle \mathbb {Q} }$ within ${\displaystyle K}$). It remains to show that if an element in ${\displaystyle K}$ which can be expressed as a sum of roots of unity in ${\displaystyle {\overline {K}}}$, then it must occur as a character value. ...

Alpha-emitters with Z ≤ 83 and half-lives lower than 10²⁰ years

No alpha decay with a half-life of at least 10²⁰ years is known, nor do I think we will be able to confirm one in this century.

A half-life of at least 10¹³ years would only have theoretical values, meaning that the alpha decay of nuclides from ¹⁸⁴Os on can be ignored. Nuclides with half-lives not exceeding that of ¹⁴⁶Sm are too short-lived to exist in nature. So ¹⁴⁷Sm (1.066×10¹¹ years) and ¹⁹⁰Pt (4.83×10¹¹ years) are the only primordial nuclides with Z ≤ 83 that emit a noticeable amount of alpha particles (still insignificant compared to ²³²Th, ²³⁵U, and ²³⁸U). In comparison, natural beta-emitters are more common - there are ⁸⁷Rb, ¹³⁸La, ¹⁷⁶Lu, ¹⁸⁷Re, and most notably, ⁴⁰K.

Nuclides with the lowest mass among isobars of mass numbers 141 ~ 209

For 141 ≤ A ≤ 209, the nuclide with the lowest mass among isobars of A (¹⁴¹Pr, ¹⁴²Nd, ¹⁴³Nd, ¹⁴⁴Nd, ¹⁴⁵Nd, ¹⁴⁶Sm, ..., ²⁰⁹Bi) are of particular interest. The following table lists their predicted half-lives here:

For nuclides in the table with predicted alpha half-lives longer than 10¹⁰⁰ years (including being stable to alpha decay), cluster decay modes (for example ¹⁶O emission) must be taken into account when modeling the half-lives. It seems that there is no concensus on predicting cluster decay half-lives, in contrast to alpha decay.

Beta-stable nuclides around technetium and promethium

It is well-known that Tc and Pm have no beta-stable isotopes, which is to say, all their isotopes undergo beta decays to form adjecant elements. Let's draw the table of beta-stable nuclides around Tc (which have an even number of protons and 54, 55, or 56 neutrons), and those around Pm (which have an even number of protons and 84, 85, or 86 neutrons):

Beta-stable nuclides around Tc (actually ⁹⁶Zr is not stable to ${\displaystyle \beta ^{-}}$)

	Z = 40	Z = 42	Z = 44	Z = 46
N = 54	94Zr	96Mo	98Ru
N = 55		97Mo	99Ru
N = 56	96Zr	98Mo	100Ru	102Pd

Beta-stable nuclides around Pm (actually ¹⁴⁸Gd is not stable to ${\displaystyle \beta ^{+}}$)

	Z = 58	Z = 60	Z = 62	Z = 64
N = 84	142Ce	144Nd	146Sm	148Gd
N = 85		145Nd	147Sm
N = 86		146Nd	148Sm	150Gd

There are some patterns of symmetry here. ⁹⁶Zr is energetically unstable to ${\displaystyle \beta ^{-}}$, but its half-life is too long (> 10²⁰ years) to be measured. The nuclide lying at the opposite corner of the second table, ¹⁴⁸Gd, has the same property: is energetically unstable to ${\displaystyle \beta ^{+}}$, but the half-life of this process (which has a spin change of ${\displaystyle 5^{-}}$) is way too long. (Note that ¹⁴⁸Gd does dacay via alpha emission to ¹⁴⁴Sm, having a short half-life of only 86.9 years. Its decay releases an energy of 3271.21 keV. For the sake of symmetry let's imagine ⁹⁶Zr absorbing an alpha particle to form ¹⁰⁰Mo; this process would release an energy of 3179.1 keV).

The nuclides in bold have the lowest energy among its isobars. There is one breaking of symmetry: the nuclides with lowest energy for A = 98 and A = 146 do not lie at opposite positions of the two tables. Yet, the energy differences of ⁹⁸Mo/⁹⁸Ru (112.75 keV) and ¹⁴⁶Nd/¹⁴⁶Sm (70.83 keV) are small. (Although those of ¹⁵²Sm/¹⁵²Gd (54.16 keV) and ¹⁶⁴Dy/¹⁶⁴Er (23.33 keV) are smaller).

The nuclide ⁹⁴Zr is stable to ${\displaystyle \beta ^{-}}$ decay by 901.7 keV, meaning that it is 901.7 keV lower in energy than ⁹⁴Nb. Likewise, the corresponding nuclide lying at the opposite corner of the second table, ¹⁵⁰Gd, is stable to ${\displaystyle \beta ^{+}}$ decay by 972 keV, meaning that it is 972 keV lower in energy than ¹⁵⁰Eu.

Factoring rational primes on the quadratic number field with discriminant ${\displaystyle D}$

• D = -376. Decomposing: A191056; remaining inert: A191086.
• D = -344. Decomposing: A191051; remaining inert: A191083.
• D = -312. Decomposing: A191047; remaining inert: A191080.
• D = -280. Decomposing: A191043; remaining inert: A191078.
• D = -248. Decomposing: A191040; remaining inert: A191076.
• D = -184. Decomposing: A191032; remaining inert: A191071.
• D = -163. Decomposing: A296921; remaining inert: A296915; not remaining inert: A257362.
• D = -152. Decomposing: A191028; remaining inert: A191069.
• D = -120. Decomposing: A191023; remaining inert: A191066.
• D = -95. Decomposing: A191057; remaining inert: A191087.
• D = -91. Decomposing: A191054; remaining inert: A191085.
• D = -88. Decomposing: A191020; remaining inert: A191064.
• D = -87. Decomposing: A191052; remaining inert: A191084.
• D = -83. Decomposing: A191050; remaining inert: A191082.
• D = -79. Decomposing: A191048; remaining inert: A191081.
• D = -71. Decomposing: A191044; remaining inert: A191079.
• D = -68. Decomposing: A296929; remaining inert: A296930; not remaining inert: A296931.
• D = -67. Decomposing: A191041; remaining inert: A191077; not remaining inert: A106933.
• D = -59. Decomposing: A191038; remaining inert: A191075.
• D = -56. Decomposing: A191017; remaining inert: A191061; not decomposing: A274504.
• D = -55. Decomposing: A191036; remaining inert: A191074.
• D = -52. Decomposing: A296926; remaining inert: A296927; not remaining inert: A296928 U {2}.
• D = -51. Decomposing: A191034; remaining inert: A191073.
• D = -47. Decomposing: A191033; remaining inert: A191072.
• D = -43. Decomposing: A191031; remaining inert: A184902; not remaining inert: A106891.
• D = -40. Decomposing: A155488; remaining inert: A296925; not remaining inert: A293859.
• D = -39. Decomposing: A191029; remaining inert: A191070.
• D = -35. Decomposing: A191026; remaining inert: A191068.
• D = -31. Decomposing: A191024; remaining inert: A191067.
• D = -24. Decomposing: A157437; remaining inert: A191059; not remaining inert: A296924.
• D = -23. Decomposing: A191021; remaining inert: A191065; not remaining inert: A296932.
• D = -20. Decomposing: A139513; remaining inert: A003626; not remaining inert: A240920 = A296922 U {2}; not decomposing: A296923 U {5}.
• D = -19. Decomposing: A191019; remaining inert: A191063; not remaining inert: A106863.
• D = -15. Decomposing: A191018; remaining inert: A191062.
• D = -11. Decomposing: A296920; remaining inert: A191060; not remaining inert: A056874.
• D = -8. Decomposing: A033200; remaining inert: A003628; not remaining inert: A033203; not decomposing: A045355.
• D = -7. Decomposing: A045386; remaining inert: A003625; not remaining inert: A045373; not decomposing: A045399.
• D = -4. Decomposing: A002144; remaining inert: A002145; not remaining inert: A002313; not decomposing: A045326.
• D = -3. Decomposing: A002476; remaining inert: A003627 = A007528 U {2}; not remaining inert: A007645; not decomposing: A045309 = A045410 U {2}.
• D = 5. Decomposing: A045468 = A064739 \ {2}; remaining inert: A003631 = A097957 U {2}; not remaining inert: A038872; not decomposing: A042993.
• D = 8. Decomposing: A001132 = A097958 \ {3}; remaining inert: A003629; not remaining inert: A038873; not decomposing: A042999.
• D = 12. Decomposing: A097933; remaining inert: A003630; not remaining inert: A038874 = A296933 U {2}; not decomposing: A038875 U {3}.
• D = 13. Decomposing: A296937; remaining inert: A038884; not remaining inert: A038883; not decomposing: A120330.
• D = 17. Decomposing: A296938; remaining inert: A038890; not remaining inert: A038889.
• D = 21. Remaining inert: A038894; not remaining inert: A038893.
• D = 24. Decomposing: A097934; remaining inert: A038877; not remaining inert: A038876.
• D = 28. Decomposing: A296934; remaining inert: A003632; not remaining inert: A038878.
• D = 29. Decomposing: A191022; remaining inert: A038902; not remaining inert: A038901.
• D = 33. Remaining inert: A038908; not remaining inert: A038907.
• D = 37. Decomposing: A191027; remaining inert: A038914; not remaining inert: A038913.
• D = 40. Decomposing: A097955; remaining inert: A038880; not remaining inert: A038879.
• D = 41. Decomposing: A191030; remaining inert: A038920; not remaining inert: A038919.
• D = 44. Decomposing: A296935; remaining inert: A296936; not remaining inert: A038881 U {2}; not decomposing: A038882 U {11}.
• D = 53. Decomposing: A191035; remaining inert: A038932; not remaining inert: A038931.
• D = 56. Remaining inert: A038886; not remaining inert: A038885.
• D = 57. Remaining inert: A038936; not remaining inert: A038935.
• D = 60. Decomposing: A097956; remaining inert: A038888; not remaining inert: A038887.
• D = 61. Decomposing: A191039; remaining inert: A038942; not remaining inert: A038941.
• D = 65. Remaining inert: A038946; not remaining inert: A038945.
• D = 69. Decomposing: A191042; remaining inert: A038952; not remaining inert: A038951.
• D = 73. Decomposing: A191045; remaining inert: A038958; not remaining inert: A038957.
• D = 76. Decomposing: A297175; remaining inert: A297176 = A038892 \ {2}; not remaining inert: A038891 U {2}.
• D = 77. Remaining inert: A038962; not remaining inert: A038961.
• D = 85. Remaining inert: A038972; not remaining inert: A038971.
• D = 88. Remaining inert: A038896; not remaining inert: A038895.
• D = 89. Decomposing: A191053; remaining inert: A038978; not remaining inert: A038977.
• D = 92. Decomposing: A297177; remaining inert: A038898; not remaining inert: A038897.
• D = 93. Decomposing: A191055; remaining inert: A038982; not remaining inert: A038981.
• D = 97. Decomposing: A191058; remaining inert: A038988; not remaining inert: A038987.
• D = 104. Remaining inert: A038900; not remaining inert: A038899.
• D = 120. Decomposing: A097959; remaining inert: A038904; not remaining inert: A038903.
• D = 124. Remaining inert: A038906; not remaining inert: A038905.
• D = 136. Decomposing: A191025; remaining inert: A038910; not remaining inert: A038909.
• D = 140. Remaining inert: A038912 \ {2}; not remaining inert: A038911 U {2}.
• D = 152. Remaining inert: A038916; not remaining inert: A038915.
• D = 156. Remaining inert: A038918; not remaining inert: A038917.
• D = 168. Remaining inert: A038922; not remaining inert: A038921.
• D = 172. Remaining inert: A038924 \ {2}; not remaining inert: A038923 U {2}.
• D = 184. Remaining inert: A038926; not remaining inert: A038925.
• D = 188. Remaining inert: A038928; not remaining inert: A038927.
• D = 204. Remaining inert: A038930 \ {2}; not remaining inert: A038929 U {2}.
• D = 220. Remaining inert: A038934; not remaining inert: A038933.
• D = 232. Decomposing: A191037; remaining inert: A038938; not remaining inert: A038937.
• D = 236. Remaining inert: A038940 \ {2}; not remaining inert: A038939 U {2}.
• D = 248. Remaining inert: A038944; not remaining inert: A038943.
• D = 264. Remaining inert: A038948; not remaining inert: A038947.
• D = 268. Remaining inert: A038950 \ {2}; not remaining inert: A038949 U {2}.
• D = 280. Remaining inert: A038954; not remaining inert: A038953.
• D = 284. Remaining inert: A038956; not remaining inert: A038955.
• D = 296. Decomposing: A191046; remaining inert: A038960; not remaining inert: A038959.
• D = 312. Remaining inert: A038964; not remaining inert: A038963.
• D = 316. Remaining inert: A038966; not remaining inert: A038965.
• D = 328. Decomposing: A191049; remaining inert: A038968; not remaining inert: A038967.
• D = 332. Remaining inert: A038970 \ {2}; not remaining inert: A038969 U {2}.
• D = 344. Remaining inert: A038974; not remaining inert: A038973.
• D = 348. Remaining inert: A038976; not remaining inert: A038975 U {2}.
• D = 364. Remaining inert: A038980 \ {2}; not remaining inert: A038979 U {2}.
• D = 376. Remaining inert: A038984; not remaining inert: A038983.
• D = 380. Remaining inert: A038986; not remaining inert: A038985.