Monthly Archives: August 2025

Theorem (square-gcd count, closed form)

Posted on August 26, 2025 by nathanalbright

For a positive integer n, let A_2(n):=\#\{\,1\le k\le n:\ \gcd(k,n)\ \text{is a perfect square}\,\}. Then A_2(n)\;=\;\sum_{d^2\mid n}\varphi\!\left(\frac{n}{d^2}\right) \;=\;\prod_{p^{\alpha}\parallel n}\frac{p^{\alpha+1}-(-1)^{\alpha+1}}{p+1}, where \varphi is Euler’s totient function and the product runs over the prime powers p^\alpha dividing n exactly. Explanation. We are … Continue reading →

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White Paper: A Typology of Musicians by Approach to Playing and Making Music

Posted on August 26, 2025 by nathanalbright

Executive Summary Musicians can be categorized not only by genre, instrument, or technical skill, but also by their approach to playing and making music. This white paper presents a typology of musician archetypes, examining the motivations, methods, and cultural roles … Continue reading →

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White Paper: Self-Help, Laws of Success, and Occult Thinking: Continuities, Crossovers, and Cultural Implications

Posted on August 25, 2025 by nathanalbright

Executive Summary The global self-help movement and the modern “laws of success” genre have deep conceptual and historical connections with occult traditions. From New Thought philosophy in the 19th century to contemporary motivational speakers, themes such as positive thinking, visualization, … Continue reading →

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White Paper: The Negative Effects of the Prosperity Gospel on Those Who Fail to Prosper

Posted on August 25, 2025 by nathanalbright

Executive Summary The prosperity gospel, also known as “Word of Faith” or “health and wealth” theology, teaches that faith, positive confession, and generous giving to religious leaders will inevitably lead to material wealth, physical health, and personal success. While appealing … Continue reading →

Posted in Christianity, Musings | Tagged , , | 2 Comments

White Paper: Institutional Messaging, Trust, and Congregational Response

Posted on August 24, 2025 by nathanalbright

Executive Summary A church-wide message was distributed across all congregations, framed around the theme of trusting God. While the primary message appeared routine and uncontroversial, the inclusion of offhand remarks about trusting authority figures, regarding them as reliable sources of … Continue reading →

Posted in Christianity, Church of God, Musings | Tagged , , , , , | 5 Comments

White Paper: Testing the Reliability of Information in a Contemporary Epistemological Landscape: A Skeptical Approach

Posted on August 24, 2025 by nathanalbright

Executive Summary In an age of information abundance, the reliability of knowledge claims is more contested than ever. While consensus and institutional authority have traditionally functioned as markers of reliability, today they often conceal ideological capture, groupthink, and incentives that … Continue reading →

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White Paper: The Additional Layers of Meaning in Institution-Wide Messaging: Communicative Dynamics of Unified Sermons and Distributed Messages

Posted on August 23, 2025 by nathanalbright

Introduction When an institution chooses to deliver a single, coordinated message across all its branches or congregations, it is doing far more than conveying content. Such an act carries with it multiple communicative layers: theological, organizational, cultural, and symbolic. In … Continue reading →

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White Paper: On the Duty to Explain One’s Faith and the Limits of Answering Critics

Posted on August 23, 2025 by nathanalbright

Introduction Christians are often confronted with accusations against their beliefs, practices, or character. Some accusers may cite 1 Peter 3:15–16, where believers are urged to “be ready always to give an answer to every man that asketh you a reason … Continue reading →

Posted in Bible, Christianity, Church of God, Musings | Tagged , , , , , | 2 Comments

White Paper: Barriers to Applying the Principle of “Honey over Vinegar” in Tone Policing

Posted on August 22, 2025 by nathanalbright

Executive Summary The proverb “you catch more flies with honey than with vinegar” reflects a longstanding recognition that persuasion and cooperation are more effectively secured through kindness than through hostility. Yet, in practice, individuals often default to harshness, defensiveness, or … Continue reading →

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White Paper: Contexts and Typology of the Biblical Calls to Not Fear and Be of Good Courage

Posted on August 22, 2025 by nathanalbright

Executive Summary The biblical commands “do not fear” and “be strong and of good courage” are among the most repeated exhortations in Scripture. They serve as reassurance in times of uncertainty, grounding faith communities in God’s promises, presence, and power. … Continue reading →

Posted in Bible, Biblical History, Christianity, History, Musings | Tagged , , , , | 1 Comment