1) The document discusses representations of prime numbers as sums of squares and relates them to modular forms. 2) Several formulas are presented for expressing prime numbers as sums of squares of integers, such as p = X^2 + 7Y^2. 3) Properties of modular forms are summarized, including representations as Fourier series and transformations under substitutions.

1. @tsujimotter
2016.5.7 13

2. LT

3. お約束
以降， は素数を表す記号とするp

4. まとめると
p = 7 or 1, 2, 4 + 7n () p = X2
+ 7Y2
p = 7 or 1, 2, 4 + 7n () p = X2
+ 7Y2
p = 5 or 1, 9 + 20n () p = X2
+ 5Y2
p = 5 or 1, 9 + 20n () p = X2
+ 5Y2
p = 2 or 1 + 4n () p = X2
+ Y2
p = 2 or 1 + 4n () p = X2
+ Y2
p = 3 or 1 + 3n () p = X2
+ 3Y2
p = 3 or 1 + 3n () p = X2
+ 3Y2
m

5. 59 = (-7)2
+ (-7) · 2 + 6 · 22
101 = (-5)2
+ (-5) · 4 + 6 · 42

6. p = X2
+ XY + 6Y2
p =
✓
X + Y
1 +
p
-23
2
◆ ✓
X + Y
1 -
p
-23
2
◆

7. f(q) = q
1Y
n=1
(1 - qn
)(1 - q23n
)
f(q) =
1X
n=1
anqn
ap = 2 () p = X2
+ XY + 6Y2

9. f k
z ! i1
a, b, c, d 2 Z, ad - bc = 1
f
✓
az + b
cz + d
◆
= (cz + d)k
f(z)

10. 保型形式のフーリエ係数
f(z) = a0 + a1q + a2q2
+ a3q3
+ · · ·

11. •
https://www.math.kyoto-u.ac.jp/insei/proceeding/2010/ito.pdf
• tsujimotter XX + XY + 6YY
http://tsujimotter.hatenablog.com/entry/6xx-xy-yy-2