Viktor Futó · Stay human
@optimista.bsky.social
Spatially written math & proofs with LaTeX - brainec.com
Alumni: University of Coimbra · Charles University in Prague
Alumni: University of Coimbra · Charles University in Prague
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐒𝐞𝐜𝐨𝐧𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
October 17, 2025 at 12:36 PM
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐒𝐞𝐜𝐨𝐧𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
In 1737 Euler showed that Σ 1/𝑝 diverges.
In 1874 Mertens’ Second Theorem estimated how quickly.
The growth is very slow: log log 𝑥.
For 𝑥 ∈ ℝ, 𝑝 prime and 𝑥 ≥ 2:
Σₚ≤𝑥 1/𝑝 = log log 𝑥 + 𝑀 + O(1/log 𝑥)
Meissel-Mertens constant
𝑀 ≈ 0.26149721…
#Math #NumberTheory
𝐄𝐮𝐥𝐞𝐫'𝐬 𝐓𝐨𝐭𝐢𝐞𝐧𝐭 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
October 16, 2025 at 12:20 PM
𝐄𝐮𝐥𝐞𝐫'𝐬 𝐓𝐨𝐭𝐢𝐞𝐧𝐭 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
φ(𝑛) = how many positive integers ≤ than 𝑛 are relatively prime to 𝑛
φ(𝑛) = |{ 𝑘 ∈ ℕ : 𝑘 ≤ 𝑛, gcd(𝑘,𝑛)=1 }|
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑙𝑦 𝑝𝑟𝑖𝑚𝑒 ≡ 𝑐𝑜𝑝𝑟𝑖𝑚𝑒 ≡ gcd(𝑎,𝑏)=1
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
φ(9) = 6
{1,2,4,5,7,8}
#math #NumberTheory
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐓𝐡𝐢𝐫𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
October 14, 2025 at 12:23 PM
𝐌𝐞𝐫𝐭𝐞𝐧𝐬' 𝐓𝐡𝐢𝐫𝐝 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
Precursor of the Prime Number Theorem
For 𝑥 ∈ ℝ and 𝑥 ≥ 2:
∏ₚ≤𝑥 (1 − 1/𝑝) = e⁻ᵞ(1+o(1))/ln𝑥
Euler–Mascheroni constant
γ ≈ 0.57721566…
#math #NumberTheory
𝐌𝐞𝐫𝐜𝐚𝐭𝐨𝐫 𝐒𝐞𝐫𝐢𝐞𝐬
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
October 13, 2025 at 1:09 PM
𝐌𝐞𝐫𝐜𝐚𝐭𝐨𝐫 𝐒𝐞𝐫𝐢𝐞𝐬
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
Also called = Taylor/Maclaurin expansion of ln(1+𝑥)
If |𝑥|<1 then:
ln(1+𝑥) = Σₙ₌₁ (−1)ⁿ⁺¹ 𝑥ⁿ/𝑛
ln(1+𝑥) = 𝑥 − 𝑥²⁄2 + 𝑥³⁄3 − 𝑥⁴⁄4 + …
#mathematics #RealAnalysis
𝐄𝐮𝐥𝐞𝐫 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐟𝐨𝐫 𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭 𝐒𝐞𝐫𝐢𝐞𝐬
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
October 10, 2025 at 1:04 PM
𝐄𝐮𝐥𝐞𝐫 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐟𝐨𝐫 𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭 𝐒𝐞𝐫𝐢𝐞𝐬
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
Core identity in analytic number theory linking Dirichlet series to primes
If 𝑓 is multiplicative & series converges absolutely at 𝑠:
∑ₙ 𝑓(𝑛)/𝑛ˢ = ∏ₚ ( ∑ₖ 𝑓(𝑝ᵏ)/𝑝ᵏˢ )
“sum over all integers” = “product over primes” (one sum per prime)
#mathematics #NumberTheory
𝐀𝐛𝐞𝐥’𝐬 𝐒𝐮𝐦𝐦𝐚𝐭𝐢𝐨𝐧 𝐅𝐨𝐫𝐦𝐮𝐥𝐚 | #NumberTheory
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
October 9, 2025 at 12:46 PM
𝐀𝐛𝐞𝐥’𝐬 𝐒𝐮𝐦𝐦𝐚𝐭𝐢𝐨𝐧 𝐅𝐨𝐫𝐦𝐮𝐥𝐚 | #NumberTheory
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
shows how a weighted sum (by 𝑓) can be expressed using cumulative sum (𝐴(𝑥)) and the way function 𝑓 changes
If 𝑓:[1,𝑁]⟼ℂ continuously differentiable & 𝐴(𝑥) = Σₙ≤𝑥 𝑎ₙ:
Σₙ₌₁ᴺ 𝑎ₙ 𝑓(𝑛) = 𝐴(𝑁) 𝑓(𝑁) − ∫₁ᴺ 𝐴(𝑥) 𝑓′(𝑥) d𝑥
Used for: ↓
𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐢𝐜𝐚𝐭𝐢𝐯𝐞 𝐈𝐧𝐝𝐢𝐜𝐚𝐭𝐨𝐫 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑜𝑟 - returns either 0 or 1
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 - its arguments are natural numbers
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 - arithmetic function for which:
𝑓(1) = 1
gcd(𝑎,𝑏) = 1 ⇒ 𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑜𝑟 - returns either 0 or 1
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 - its arguments are natural numbers
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 - arithmetic function for which:
𝑓(1) = 1
gcd(𝑎,𝑏) = 1 ⇒ 𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)
October 7, 2025 at 12:41 PM
𝐌𝐮𝐥𝐭𝐢𝐩𝐥𝐢𝐜𝐚𝐭𝐢𝐯𝐞 𝐈𝐧𝐝𝐢𝐜𝐚𝐭𝐨𝐫 𝐅𝐮𝐧𝐜𝐭𝐢𝐨𝐧
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑜𝑟 - returns either 0 or 1
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 - its arguments are natural numbers
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 - arithmetic function for which:
𝑓(1) = 1
gcd(𝑎,𝑏) = 1 ⇒ 𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)
𝑖𝑛𝑑𝑖𝑐𝑎𝑡𝑜𝑟 - returns either 0 or 1
𝑎𝑟𝑖𝑡ℎ𝑚𝑒𝑡𝑖𝑐 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 - its arguments are natural numbers
𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑣𝑒 - arithmetic function for which:
𝑓(1) = 1
gcd(𝑎,𝑏) = 1 ⇒ 𝑓(𝑎𝑏) = 𝑓(𝑎)𝑓(𝑏)
𝐌𝐨𝐝𝐮𝐥𝐚𝐫 𝐈𝐧𝐯𝐞𝐫𝐬𝐞
(= “modular multiplicative inverse”)
Let 𝑎 ∈ ℤ and 𝑥 ∈ ℤ₊ with gcd(𝑎,𝑥) = 1.
Modular inverse of 𝑎 modulo 𝑥 is each 𝑏 for which:
𝑎𝑏 ≡ 1 (mod 𝑥)
𝑭𝒐𝒓 𝒆𝒙𝒂𝒎𝒑𝒍𝒆:
Modular inverses of 3 mod 7 are 5,12,19,26,...
𝑁𝑜𝑡𝑖𝑐𝑒: All inverses differ by 𝑥 (here 𝑥 = 7).
#numbertheory
(= “modular multiplicative inverse”)
Let 𝑎 ∈ ℤ and 𝑥 ∈ ℤ₊ with gcd(𝑎,𝑥) = 1.
Modular inverse of 𝑎 modulo 𝑥 is each 𝑏 for which:
𝑎𝑏 ≡ 1 (mod 𝑥)
𝑭𝒐𝒓 𝒆𝒙𝒂𝒎𝒑𝒍𝒆:
Modular inverses of 3 mod 7 are 5,12,19,26,...
𝑁𝑜𝑡𝑖𝑐𝑒: All inverses differ by 𝑥 (here 𝑥 = 7).
#numbertheory
October 6, 2025 at 12:14 PM
𝐌𝐨𝐝𝐮𝐥𝐚𝐫 𝐈𝐧𝐯𝐞𝐫𝐬𝐞
(= “modular multiplicative inverse”)
Let 𝑎 ∈ ℤ and 𝑥 ∈ ℤ₊ with gcd(𝑎,𝑥) = 1.
Modular inverse of 𝑎 modulo 𝑥 is each 𝑏 for which:
𝑎𝑏 ≡ 1 (mod 𝑥)
𝑭𝒐𝒓 𝒆𝒙𝒂𝒎𝒑𝒍𝒆:
Modular inverses of 3 mod 7 are 5,12,19,26,...
𝑁𝑜𝑡𝑖𝑐𝑒: All inverses differ by 𝑥 (here 𝑥 = 7).
#numbertheory
(= “modular multiplicative inverse”)
Let 𝑎 ∈ ℤ and 𝑥 ∈ ℤ₊ with gcd(𝑎,𝑥) = 1.
Modular inverse of 𝑎 modulo 𝑥 is each 𝑏 for which:
𝑎𝑏 ≡ 1 (mod 𝑥)
𝑭𝒐𝒓 𝒆𝒙𝒂𝒎𝒑𝒍𝒆:
Modular inverses of 3 mod 7 are 5,12,19,26,...
𝑁𝑜𝑡𝑖𝑐𝑒: All inverses differ by 𝑥 (here 𝑥 = 7).
#numbertheory
𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭’𝐬 𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝐨𝐧 𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐏𝐫𝐨𝐠𝐫𝐞𝐬𝐬𝐢𝐨𝐧𝐬
Let 𝑎,𝑏 be positive integers with gcd(𝑎,𝑏)=1.
⇒ "There are infinitely many primes 𝑝 such that 𝑝 ≡ 𝑎 (mod 𝑏)."
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
For gcd(2,5) = 1, there are infinitely many primes 𝑝 ≡ 2 (mod 5): 2, 7, 17, 37, 47, 67, 97, 107,…
Let 𝑎,𝑏 be positive integers with gcd(𝑎,𝑏)=1.
⇒ "There are infinitely many primes 𝑝 such that 𝑝 ≡ 𝑎 (mod 𝑏)."
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
For gcd(2,5) = 1, there are infinitely many primes 𝑝 ≡ 2 (mod 5): 2, 7, 17, 37, 47, 67, 97, 107,…
October 3, 2025 at 12:52 PM
𝐃𝐢𝐫𝐢𝐜𝐡𝐥𝐞𝐭’𝐬 𝐓𝐡𝐞𝐨𝐫𝐞𝐦 𝐨𝐧 𝐀𝐫𝐢𝐭𝐡𝐦𝐞𝐭𝐢𝐜 𝐏𝐫𝐨𝐠𝐫𝐞𝐬𝐬𝐢𝐨𝐧𝐬
Let 𝑎,𝑏 be positive integers with gcd(𝑎,𝑏)=1.
⇒ "There are infinitely many primes 𝑝 such that 𝑝 ≡ 𝑎 (mod 𝑏)."
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
For gcd(2,5) = 1, there are infinitely many primes 𝑝 ≡ 2 (mod 5): 2, 7, 17, 37, 47, 67, 97, 107,…
Let 𝑎,𝑏 be positive integers with gcd(𝑎,𝑏)=1.
⇒ "There are infinitely many primes 𝑝 such that 𝑝 ≡ 𝑎 (mod 𝑏)."
𝑬𝒙𝒂𝒎𝒑𝒍𝒆:
For gcd(2,5) = 1, there are infinitely many primes 𝑝 ≡ 2 (mod 5): 2, 7, 17, 37, 47, 67, 97, 107,…
𝐃𝐞𝐧𝐬𝐢𝐭𝐲 𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃.𝐋.)
Let 𝐴,𝐵,𝐶 be positive integers with gcd(𝐴,𝐵)=1.
Define 𝑙(𝑥,𝑦) = 𝐴𝑥𝑦 + 𝐵𝑥 + 𝐶𝑦
⇒ "The values of 𝑙(𝑥,𝑦) are a density-one subset of the positive integers."
Leads to proof of Density-one version of the Davis–Lelièvre conjecture.
Let 𝐴,𝐵,𝐶 be positive integers with gcd(𝐴,𝐵)=1.
Define 𝑙(𝑥,𝑦) = 𝐴𝑥𝑦 + 𝐵𝑥 + 𝐶𝑦
⇒ "The values of 𝑙(𝑥,𝑦) are a density-one subset of the positive integers."
Leads to proof of Density-one version of the Davis–Lelièvre conjecture.
October 2, 2025 at 12:53 PM
𝐃𝐞𝐧𝐬𝐢𝐭𝐲 𝐋𝐞𝐦𝐦𝐚 𝟒.𝟒. (𝐃.𝐋.)
Let 𝐴,𝐵,𝐶 be positive integers with gcd(𝐴,𝐵)=1.
Define 𝑙(𝑥,𝑦) = 𝐴𝑥𝑦 + 𝐵𝑥 + 𝐶𝑦
⇒ "The values of 𝑙(𝑥,𝑦) are a density-one subset of the positive integers."
Leads to proof of Density-one version of the Davis–Lelièvre conjecture.
Let 𝐴,𝐵,𝐶 be positive integers with gcd(𝐴,𝐵)=1.
Define 𝑙(𝑥,𝑦) = 𝐴𝑥𝑦 + 𝐵𝑥 + 𝐶𝑦
⇒ "The values of 𝑙(𝑥,𝑦) are a density-one subset of the positive integers."
Leads to proof of Density-one version of the Davis–Lelièvre conjecture.
𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞
“Davis–Lelièvre Conjecture is true/1 in density”
𝐷𝑎𝑣𝑖𝑠–𝐿𝑒𝑙𝑖𝑒̀𝑣𝑟𝑒 𝐶𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 – see previous post
𝑡𝑟𝑢𝑒 𝑖𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 – the ratio of even integers where the conjecture holds to all even integers converges → 1
...
“Davis–Lelièvre Conjecture is true/1 in density”
𝐷𝑎𝑣𝑖𝑠–𝐿𝑒𝑙𝑖𝑒̀𝑣𝑟𝑒 𝐶𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 – see previous post
𝑡𝑟𝑢𝑒 𝑖𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 – the ratio of even integers where the conjecture holds to all even integers converges → 1
...
September 30, 2025 at 1:50 PM
𝐃𝐞𝐧𝐬𝐢𝐭𝐲-𝐨𝐧𝐞 𝐃𝐚𝐯𝐢𝐬–𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞
“Davis–Lelièvre Conjecture is true/1 in density”
𝐷𝑎𝑣𝑖𝑠–𝐿𝑒𝑙𝑖𝑒̀𝑣𝑟𝑒 𝐶𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 – see previous post
𝑡𝑟𝑢𝑒 𝑖𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 – the ratio of even integers where the conjecture holds to all even integers converges → 1
...
“Davis–Lelièvre Conjecture is true/1 in density”
𝐷𝑎𝑣𝑖𝑠–𝐿𝑒𝑙𝑖𝑒̀𝑣𝑟𝑒 𝐶𝑜𝑛𝑗𝑒𝑐𝑡𝑢𝑟𝑒 – see previous post
𝑡𝑟𝑢𝑒 𝑖𝑛 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 – the ratio of even integers where the conjecture holds to all even integers converges → 1
...
𝐃𝐚𝐯𝐢𝐬-𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞 𝐂𝐨𝐧𝐣𝐞𝐜𝐭𝐮𝐫𝐞
"Every positive even integer except 2,12,14 and 18 occurs."
𝑜𝑐𝑐𝑢𝑟𝑠 - ∃ an asymmetric closed billiard trajectory on the regular pentagon whose combinatorial period length equals that integer
"Every positive even integer except 2,12,14 and 18 occurs."
𝑜𝑐𝑐𝑢𝑟𝑠 - ∃ an asymmetric closed billiard trajectory on the regular pentagon whose combinatorial period length equals that integer
September 29, 2025 at 2:33 PM
𝐃𝐚𝐯𝐢𝐬-𝐋𝐞𝐥𝐢𝐞̀𝐯𝐫𝐞 𝐂𝐨𝐧𝐣𝐞𝐜𝐭𝐮𝐫𝐞
"Every positive even integer except 2,12,14 and 18 occurs."
𝑜𝑐𝑐𝑢𝑟𝑠 - ∃ an asymmetric closed billiard trajectory on the regular pentagon whose combinatorial period length equals that integer
"Every positive even integer except 2,12,14 and 18 occurs."
𝑜𝑐𝑐𝑢𝑟𝑠 - ∃ an asymmetric closed billiard trajectory on the regular pentagon whose combinatorial period length equals that integer
𝐅𝐮𝐧𝐝𝐚𝐦𝐞𝐧𝐭𝐚𝐥 𝐋𝐞𝐦𝐦𝐚 𝐨𝐟 𝐕𝐚𝐫𝐢𝐚𝐭𝐢𝐨𝐧𝐚𝐥 𝐂𝐚𝐥𝐜𝐮𝐥𝐮𝐬
In variational calculus, when we search for an extremum of a functional, we often end up with an integral of the form:
∫ 𝑓(x) φ(x) dx = 0
If 𝑓 is locally integrable & φ is “nice” function
*[nice = smooth, compact support]...
In variational calculus, when we search for an extremum of a functional, we often end up with an integral of the form:
∫ 𝑓(x) φ(x) dx = 0
If 𝑓 is locally integrable & φ is “nice” function
*[nice = smooth, compact support]...
September 26, 2025 at 2:09 PM
𝐅𝐮𝐧𝐝𝐚𝐦𝐞𝐧𝐭𝐚𝐥 𝐋𝐞𝐦𝐦𝐚 𝐨𝐟 𝐕𝐚𝐫𝐢𝐚𝐭𝐢𝐨𝐧𝐚𝐥 𝐂𝐚𝐥𝐜𝐮𝐥𝐮𝐬
In variational calculus, when we search for an extremum of a functional, we often end up with an integral of the form:
∫ 𝑓(x) φ(x) dx = 0
If 𝑓 is locally integrable & φ is “nice” function
*[nice = smooth, compact support]...
In variational calculus, when we search for an extremum of a functional, we often end up with an integral of the form:
∫ 𝑓(x) φ(x) dx = 0
If 𝑓 is locally integrable & φ is “nice” function
*[nice = smooth, compact support]...
𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
“The sum of all flows inside a volume equals the sum of flows across its boundary.”
Cut any shape into two parts. The net flow across the cut from one side is the opposite of the other, so they cancel. The boundary sum remains the same. #math
“The sum of all flows inside a volume equals the sum of flows across its boundary.”
Cut any shape into two parts. The net flow across the cut from one side is the opposite of the other, so they cancel. The boundary sum remains the same. #math
September 23, 2025 at 1:50 PM
𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐓𝐡𝐞𝐨𝐫𝐞𝐦
“The sum of all flows inside a volume equals the sum of flows across its boundary.”
Cut any shape into two parts. The net flow across the cut from one side is the opposite of the other, so they cancel. The boundary sum remains the same. #math
“The sum of all flows inside a volume equals the sum of flows across its boundary.”
Cut any shape into two parts. The net flow across the cut from one side is the opposite of the other, so they cancel. The boundary sum remains the same. #math
𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐑𝐮𝐥𝐞
multivariable analogue to ordinary product rule
d(𝑓𝑔)/dx = (d𝑓/dx)𝑔 + 𝑓(d𝑔/dx)
∇⋅(𝜙𝐅) = (∇𝜙)⋅𝐅 + 𝜙(∇⋅𝐅)
𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫𝐬
d/dx - one-variable | ∇ - multivariable
e.g. ∇ = (∂/∂x,∂/∂y,∂/∂z) [3 variables]
multivariable analogue to ordinary product rule
d(𝑓𝑔)/dx = (d𝑓/dx)𝑔 + 𝑓(d𝑔/dx)
∇⋅(𝜙𝐅) = (∇𝜙)⋅𝐅 + 𝜙(∇⋅𝐅)
𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫𝐬
d/dx - one-variable | ∇ - multivariable
e.g. ∇ = (∂/∂x,∂/∂y,∂/∂z) [3 variables]
September 19, 2025 at 3:11 PM
𝐃𝐢𝐯𝐞𝐫𝐠𝐞𝐧𝐜𝐞 𝐏𝐫𝐨𝐝𝐮𝐜𝐭 𝐑𝐮𝐥𝐞
multivariable analogue to ordinary product rule
d(𝑓𝑔)/dx = (d𝑓/dx)𝑔 + 𝑓(d𝑔/dx)
∇⋅(𝜙𝐅) = (∇𝜙)⋅𝐅 + 𝜙(∇⋅𝐅)
𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫𝐬
d/dx - one-variable | ∇ - multivariable
e.g. ∇ = (∂/∂x,∂/∂y,∂/∂z) [3 variables]
multivariable analogue to ordinary product rule
d(𝑓𝑔)/dx = (d𝑓/dx)𝑔 + 𝑓(d𝑔/dx)
∇⋅(𝜙𝐅) = (∇𝜙)⋅𝐅 + 𝜙(∇⋅𝐅)
𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐨𝐩𝐞𝐫𝐚𝐭𝐨𝐫𝐬
d/dx - one-variable | ∇ - multivariable
e.g. ∇ = (∂/∂x,∂/∂y,∂/∂z) [3 variables]
Gateaux Derivative
– generalizes the derivative to functionals
Ordinary derivative → 𝑓(𝑥+ℎ)
Functional derivative → 𝑓(𝑢+𝜖𝜙)
𝜙 - test function
𝜖𝜙 - perturbation (𝜖 → 0)
If 𝑢,𝜙 = functions & 𝓕 = functional
⇒ Gateaux derivative = Functional derivative
Gateaux ⊃ Functional ⊃ Ordinary
– generalizes the derivative to functionals
Ordinary derivative → 𝑓(𝑥+ℎ)
Functional derivative → 𝑓(𝑢+𝜖𝜙)
𝜙 - test function
𝜖𝜙 - perturbation (𝜖 → 0)
If 𝑢,𝜙 = functions & 𝓕 = functional
⇒ Gateaux derivative = Functional derivative
Gateaux ⊃ Functional ⊃ Ordinary
September 18, 2025 at 2:32 PM
Gateaux Derivative
– generalizes the derivative to functionals
Ordinary derivative → 𝑓(𝑥+ℎ)
Functional derivative → 𝑓(𝑢+𝜖𝜙)
𝜙 - test function
𝜖𝜙 - perturbation (𝜖 → 0)
If 𝑢,𝜙 = functions & 𝓕 = functional
⇒ Gateaux derivative = Functional derivative
Gateaux ⊃ Functional ⊃ Ordinary
– generalizes the derivative to functionals
Ordinary derivative → 𝑓(𝑥+ℎ)
Functional derivative → 𝑓(𝑢+𝜖𝜙)
𝜙 - test function
𝜖𝜙 - perturbation (𝜖 → 0)
If 𝑢,𝜙 = functions & 𝓕 = functional
⇒ Gateaux derivative = Functional derivative
Gateaux ⊃ Functional ⊃ Ordinary
On Euclidean spaces Ω
✓ minimizing movements are easy to approximate
On a curved surface 𝑆
✗ difficult—surface-only point interactions require a mesh
👉 We use *Closest Point Method*
For each point on 𝑆 take its normal vector. Assign to each point 𝒙 ∉ 𝑆 on the normal the same value as at its...
✓ minimizing movements are easy to approximate
On a curved surface 𝑆
✗ difficult—surface-only point interactions require a mesh
👉 We use *Closest Point Method*
For each point on 𝑆 take its normal vector. Assign to each point 𝒙 ∉ 𝑆 on the normal the same value as at its...
September 16, 2025 at 2:50 PM
On Euclidean spaces Ω
✓ minimizing movements are easy to approximate
On a curved surface 𝑆
✗ difficult—surface-only point interactions require a mesh
👉 We use *Closest Point Method*
For each point on 𝑆 take its normal vector. Assign to each point 𝒙 ∉ 𝑆 on the normal the same value as at its...
✓ minimizing movements are easy to approximate
On a curved surface 𝑆
✗ difficult—surface-only point interactions require a mesh
👉 We use *Closest Point Method*
For each point on 𝑆 take its normal vector. Assign to each point 𝒙 ∉ 𝑆 on the normal the same value as at its...
Minimizing movements
- a variational method for approximating the gradient flow by iteratively minimizing functionals 𝓕ₙ
- 𝑛 marks 𝑛-th time step in the evolution of the field 𝑢
- at each 𝑛-th time step, we find the next such 𝑢, for which 𝓕ₙ₊₁(𝑢) returns the lowest value. We call such 𝑢 minimizer
- a variational method for approximating the gradient flow by iteratively minimizing functionals 𝓕ₙ
- 𝑛 marks 𝑛-th time step in the evolution of the field 𝑢
- at each 𝑛-th time step, we find the next such 𝑢, for which 𝓕ₙ₊₁(𝑢) returns the lowest value. We call such 𝑢 minimizer
September 15, 2025 at 2:14 PM
Minimizing movements
- a variational method for approximating the gradient flow by iteratively minimizing functionals 𝓕ₙ
- 𝑛 marks 𝑛-th time step in the evolution of the field 𝑢
- at each 𝑛-th time step, we find the next such 𝑢, for which 𝓕ₙ₊₁(𝑢) returns the lowest value. We call such 𝑢 minimizer
- a variational method for approximating the gradient flow by iteratively minimizing functionals 𝓕ₙ
- 𝑛 marks 𝑛-th time step in the evolution of the field 𝑢
- at each 𝑛-th time step, we find the next such 𝑢, for which 𝓕ₙ₊₁(𝑢) returns the lowest value. We call such 𝑢 minimizer
In light of Math, Inc.’s success on the autoformalization of the strong Prime Number Theorem, I thought some might find useful my spatial rewriting of Newman’s proof of the weak version of Prime Number Theorem.
Could spatial story format make AI-generated formalizations human-readable for us all?
Could spatial story format make AI-generated formalizations human-readable for us all?
September 12, 2025 at 5:26 PM
In light of Math, Inc.’s success on the autoformalization of the strong Prime Number Theorem, I thought some might find useful my spatial rewriting of Newman’s proof of the weak version of Prime Number Theorem.
Could spatial story format make AI-generated formalizations human-readable for us all?
Could spatial story format make AI-generated formalizations human-readable for us all?
Euler-Lagrange equation(s)
- let us find stationary points of a functional ℱ
- for a function we set 𝑓'(𝑥)=0 to find extrema
- for a functional, Euler-Lagrange ~ “derivative for functionals”
The form can vary depending on the model and terms we use. Here we seek such 𝑢 for which ℱ attains extremum
- let us find stationary points of a functional ℱ
- for a function we set 𝑓'(𝑥)=0 to find extrema
- for a functional, Euler-Lagrange ~ “derivative for functionals”
The form can vary depending on the model and terms we use. Here we seek such 𝑢 for which ℱ attains extremum
September 12, 2025 at 2:08 PM
Euler-Lagrange equation(s)
- let us find stationary points of a functional ℱ
- for a function we set 𝑓'(𝑥)=0 to find extrema
- for a functional, Euler-Lagrange ~ “derivative for functionals”
The form can vary depending on the model and terms we use. Here we seek such 𝑢 for which ℱ attains extremum
- let us find stationary points of a functional ℱ
- for a function we set 𝑓'(𝑥)=0 to find extrema
- for a functional, Euler-Lagrange ~ “derivative for functionals”
The form can vary depending on the model and terms we use. Here we seek such 𝑢 for which ℱ attains extremum
Lagrangian density ("Lagrangian") 𝐿/ℒ
- a function that describes system's dynamics
- arguments: position, 𝑢 at that position, 𝑢's change in time and in space.
𝑢(𝑥,𝑦,𝑧) = system's property at [x,y,z] (e.g. temperature)
Lagrangian functional ℱ
- integrates 𝐿 over the space domain Ω [x,y,z]
- a function that describes system's dynamics
- arguments: position, 𝑢 at that position, 𝑢's change in time and in space.
𝑢(𝑥,𝑦,𝑧) = system's property at [x,y,z] (e.g. temperature)
Lagrangian functional ℱ
- integrates 𝐿 over the space domain Ω [x,y,z]
September 10, 2025 at 10:31 PM
Lagrangian density ("Lagrangian") 𝐿/ℒ
- a function that describes system's dynamics
- arguments: position, 𝑢 at that position, 𝑢's change in time and in space.
𝑢(𝑥,𝑦,𝑧) = system's property at [x,y,z] (e.g. temperature)
Lagrangian functional ℱ
- integrates 𝐿 over the space domain Ω [x,y,z]
- a function that describes system's dynamics
- arguments: position, 𝑢 at that position, 𝑢's change in time and in space.
𝑢(𝑥,𝑦,𝑧) = system's property at [x,y,z] (e.g. temperature)
Lagrangian functional ℱ
- integrates 𝐿 over the space domain Ω [x,y,z]
Time discretization = evaluating system's dynamics in time-discrete steps instead of every instant
Here we apply it on the heat equation PDE and derive an approximation of a time derivative by the backward difference (𝓾 at next step − 𝓾 at current step) / 𝓱 - "backward Euler time step".
Here we apply it on the heat equation PDE and derive an approximation of a time derivative by the backward difference (𝓾 at next step − 𝓾 at current step) / 𝓱 - "backward Euler time step".
September 9, 2025 at 4:09 PM
Time discretization = evaluating system's dynamics in time-discrete steps instead of every instant
Here we apply it on the heat equation PDE and derive an approximation of a time derivative by the backward difference (𝓾 at next step − 𝓾 at current step) / 𝓱 - "backward Euler time step".
Here we apply it on the heat equation PDE and derive an approximation of a time derivative by the backward difference (𝓾 at next step − 𝓾 at current step) / 𝓱 - "backward Euler time step".
Wave Equation
- describes wave movements through space
- 𝑢 here is displacement and its second derivative = acceleration
- acc. equals 𝛼 times Laplacian 𝛥𝑢 - i.e. how much displacement 𝑢 bends from its neighbours
- describes wave movements through space
- 𝑢 here is displacement and its second derivative = acceleration
- acc. equals 𝛼 times Laplacian 𝛥𝑢 - i.e. how much displacement 𝑢 bends from its neighbours
July 21, 2025 at 4:21 PM
Wave Equation
- describes wave movements through space
- 𝑢 here is displacement and its second derivative = acceleration
- acc. equals 𝛼 times Laplacian 𝛥𝑢 - i.e. how much displacement 𝑢 bends from its neighbours
- describes wave movements through space
- 𝑢 here is displacement and its second derivative = acceleration
- acc. equals 𝛼 times Laplacian 𝛥𝑢 - i.e. how much displacement 𝑢 bends from its neighbours
when you like your lines, but you also want to distinguish your subgraph
July 7, 2025 at 3:38 PM
when you like your lines, but you also want to distinguish your subgraph