Yuuki Yamagishi

#small language model Open access Sep 2026

Mersenne Primes Are Expected Without End and Fermat Primes to Stop, the Growth of the Exponent Splitting the Sum ── correcting for not being divisible by small primes, the expected number of Mersenne primes grows by the same amount each time the exponent limit is multiplied by 10, while for Fermat numbers it is used up within the first ten or so ── the separator is whether the exponent grows one step at a time or by doubling ── [Paper 1018]

Whether infinitely many numbers 2 to the power p minus 1 (Mersenne numbers) are prime, and whether any number 2 to the power 2 to the power n plus 1 (a Fermat number) beyond F4 is prime, are both open. That there are infinitely many Mersenne primes is the same question as whether there are infinitely many even perfect...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Primitive Roots Break the Independent Estimate Only When the Square-Free Base Leaves 1 on Division by 4 ── for a = 5, being a quadratic non-residue and the remainder of p on division by 5 are tied together by reciprocity, and the share of primes with 5 as a primitive root becomes 1 + 1/19 times the independent product ── the separator is whether a leaves 1 or 3 on division by 4 ── [Paper 1020]

An integer a is a primitive root of a prime p when its powers produce every nonzero remainder on division by p. Whether infinitely many primes have a as a primitive root (Artin’s conjecture) is open for every a that is neither a square nor −1. This paper counts the share of primes for which a is a primitive root and li...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Continued-Fraction Statistics Cannot Tell Known Transcendentals from Unknown Ones or from a Cubic Irrational ── the geometric mean of partial quotients, the growth of denominators and the share of ones all fall within the scatter of random numbers for eight named numbers, and only periodic quadratic numbers and the patterned e stand out ── the separator is whether one speaks of almost all numbers in measure or of this number by name ── [Paper 1021]

Whether π + e, πe or γ is transcendental is not known. On the other hand, it is known that for almost all real numbers the geometric mean of the partial quotients of the continued fraction approaches Khinchin’s constant (Khinchin). This paper expands to 3000 terms the continued fractions of numbers known to be transcen...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Reversing an Iron Rod's Magnetization Spins It at a Rate Independent of Length, Halved When g Is 2 Instead of 1 ── the angular speed is 8 Mₛ mₑ/(g e ρ r²), inversely proportional to the square of the radius, and on a fibre of period 10 seconds one reversal swings the rod 0.900766 degrees for g equal to 1 and 0.450383 degrees for 2 ── the separator is whether the angular momentum is carried by orbital motion or by spin ── [Paper 1012]

Anything with a magnetic moment also carries angular momentum, so reversing a magnetization sets the body spinning with the difference (Einstein and de Haas 1915). This paper counts, for an iron cylinder, the angular speed gained in a reversal, the swing on a torsion fibre, and how small the rotational energy is. No ne...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Counting a Ball by Lattice Points, the Error Comes from the Skin in Low Dimensions and from One Shell in High Ones ── lining up the Gauss circle problem by dimension, the unsolved dimensions 2 and 3 are those where the accumulated fluctuation of the skin is not beaten by the jump of a single shell ── the separator is whether the error is an accumulation of jumps or a single jump ── [Paper 1015]

Counting the lattice points in a ball of radius r departs from the volume V times r to the power d (the content). Asking how large this departure is in two dimensions is the Gauss circle problem, and it is still unsolved in two and three dimensions. This paper measures the departure in the same way for dimensions d = 2...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Whether a Sum of Powers Has Many Solutions Depends on Whether Its Exponents Make a Spherical, Flat or Hyperbolic Triangle ── when the triangle with angles π/a, π/b, π/c has angle sum above π it lies on a sphere and the primitive solutions form families, below π it is hyperbolic and they are finite, and Fermat's n = 2, 3, 4 fall one into each ── the separator is whether the sum of the reciprocal exponents exceeds 1 ── [Paper 1016]

The integer solutions of x to the power a plus y to the power b equal to z to the power c (the generalized Fermat equation) change greatly with the exponents. The Beal conjecture and the Fermat-Catalan conjecture are open questions about parts of this. This paper forms from the exponents a triangle with angles π/a, π/b...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Critical Mass Is Set by Skin Balancing Content, Yet a Cube's Extra Volume Does Not Match Its Surface Coefficient ── in a bare one-group diffusion sphere, neutrons leaking through the skin over those gained in the content equal the square of the critical radius over the radius, and a cube's critical-volume factor 1.240490 is smaller than its equal-volume surface coefficient 1.240701 ── the separator is whether the weight of a shape is set by surface area or by the first eigenvalue ── [Paper 1007]

Whether a fission chain reaction keeps going is decided by the balance between neutrons gained in the content and neutrons leaking through the skin. This paper counts that balance in a bare one-group diffusion core (no reflector, no extrapolation distance, lengths measured in the diffusion length L, and the material re...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

The Angle a Foucault Pendulum Turns in a Day Is 2π Minus the Solid Angle of the Sky Above the Latitude Circle ── the angle 2pisinphi through which the plane of swing turns in one sidereal day is 2pi minus the change of direction under parallel transport around the latitude circle (the solid angle of the cap, Omega=2pi(1-sinphi)), and does not depend on the pendulum's period ── the separator is whether the day is counted in sidereal or in solar days ── [Paper 903]

The angle 2pisinphi that a Foucault pendulum turns in a day is usually taken as a number that comes out of the Coriolis force. This paper obtains the same angle in three ways: by numerical integration of the equations of motion, by parallel transport along the latitude circle, and from the solid angle of the cap above...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

If Gravity's Direction Merely Lagged at the Speed of Light, the Earth's Semi-Major Axis Would Double in 663.55 Revolutions ── tilting the direction of the Sun's pull toward the motion by the angle of aberration makes the orbit grow every revolution, with sqrt a increasing in proportion to the number of revolutions ── the separator is whether only the direction is delayed or the velocity terms are included too ── [Paper 909]

If gravity travels at a finite speed, the direction of the Sun’s pull should tilt slightly as in aberration, and it is easy to assume that the tilt is too small to change anything. This paper steps an orbit in a model where only the direction of gravity is tilted by the angle of aberration, and counts what that small t...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

The Roads to Fermat's Last Theorem Reach Different Exponents, and Descent and Kummer Together Miss 92 Exponents up to 1000 ── together the two still miss 92 exponents up to 1000 (the smallest is 37) ── only Wiles's road reached them all ── [Paper 642]

For n>=3, x^n+y^n=z^n has no solution in coprime positive integers (Fermat’s Last Theorem, proved by Wiles in 1995). This paper proves nothing; it sets out the roads that reach this “no solution” and counts which exponents each reaches. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

Carrying a Polarisation Around a Closed Path Shifts Its Phase by Half the Enclosed Solid Angle ── enclosing a hemisphere gives a phase of pi and flips the sign, the same half as the Berry phase of a qubit ── the separator is whether one measures intensity alone or the phase by interference with a reference ── [Paper 906]

Passing light through a few polarisers and back to its original polarisation, it is easy to assume the light has returned to what it was. This paper follows the state of polarisation as a point on the Poincaré sphere and compares the phase left after a closed path with the solid angle the path encloses. No new theorem...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations

#small language model Open access Sep 2026

The Wind on a Sail and the Apparent Place of a Star Are One and the Same Formula ── the formula that turns a boat-to-wind ratio into an apparent wind gives an aberration of 20.492653 arcseconds when fed the Earth's beta=9.935118x10^-5 ── the two formulas part at beta=0.5, where they differ by 3.4349 degrees ── [Paper 524]

The apparent wind of a sailing craft and the aberration of starlight are taught as separate matters. What this paper shows is that at low speed they are one and the same function, and that they part only where the rule for composing velocities changes. No new theorem or law is claimed. Scope of this paper (scope note):...

Yuuki Yamagishi · Zenodo (CERN European Organi... · 0 citations