Uniswap calculates token prices using an automated market maker formula that maintains a constant product relationship between the reserves of two tokens in a liquidity pool. The basic formula is x * y = k, where:

• x and y represent the reserves of two tokens
• k is a constant that remains unchanged after trades (excluding fees)

When a user trades token X for token Y, they add to the X reserves and remove from Y reserves. The amount of Y they receive is calculated to maintain the constant product k. This creates a price curve where larger trades cause more significant price impact, naturally simulating supply and demand mechanics. In V3, this calculation becomes more complex with the introduction of concentrated liquidity and price ranges, using square root price space for more efficient computation.

From a technical perspective, the key differences between Uniswap V2 and V3 include:

• Concentrated Liquidity: V3 implements position-specific price ranges, requiring a complete redesign of the core pool contract with non-fungible liquidity positions instead of fungible LP tokens.
• Data Structure: V3 uses a more complex tick-based data structure to track prices and liquidity at different price points, compared to V2's simpler constant product formula.
• Oracle Implementation: V3 features an enhanced oracle system with geometric mean TWAP and configurable observation arrays.
• Fee Structure: V3 implements multiple fee tiers at the factory level, requiring additional pool deployment logic and routing complexity.
• Position Management: V3 uses the ERC-721 standard for NFT position representation, necessitating a more complex position manager contract architecture.
• Math Precision: V3 implements Q64.96 fixed-point arithmetic for higher precision in price calculations, compared to V2's simpler approach.

The Uniswap app connects to the Ethereum blockchain through several technical layers:

• Web3 Provider Interface: The app uses libraries like ethers.js or web3.js to interact with Ethereum nodes.
• Wallet Connection: Integration with browser wallets (MetaMask), mobile wallets (WalletConnect), or hardware wallets via standardized interfaces.
• RPC Endpoints: The app communicates with Ethereum nodes through JSON-RPC API calls to providers like Infura, Alchemy, or self-hosted nodes.
• Transaction Management: The interface handles transaction creation, gas estimation, signature requests, and monitoring of transaction status.
• Data Indexing: For efficiency, the app queries indexed data from The Graph protocol rather than making direct blockchain calls for historical data.
• State Synchronization: The app maintains local state that mirrors relevant blockchain state, updating through event subscriptions and polling.

This multi-layered approach balances user experience with decentralization principles.

Uniswap V3's concentrated liquidity implementation solves several technical challenges:

• Capital Efficiency: V3 allows liquidity providers to focus their capital within specific price ranges, achieving up to 4000x capital efficiency compared to V2. This required developing a complex tick-based system for tracking liquidity distribution.
• Position Management: Managing non-fungible positions with unique price ranges required implementing a comprehensive position tracking system using ERC-721 tokens.
• Fee Accounting: Since LPs can have different positions at different price ranges, V3 needed to develop a per-position fee accounting system that accurately tracks and attributes fees.
• Price Range Mechanics: V3 implements "ticks" as discrete price points that define position boundaries, requiring efficient data structures to track active and inactive liquidity.
• Mathematical Precision: Operating in "sqrt price space" for computational efficiency while maintaining high precision required implementing specialized fixed-point math libraries.

These solutions collectively enable V3's superior capital efficiency while maintaining the core AMM model.

The Uniswap wallet ensures security of user funds through multiple technical safeguards:

• Non-custodial Architecture: The wallet never takes custody of private keys or funds—all cryptographic operations happen locally on the user's device.
• Secure Key Management: Private keys are encrypted with industry-standard algorithms and protected by biometric or PIN authentication.
• Transaction Safeguards: All transactions are simulated before submission to prevent unexpected outcomes, with clear confirmations for approvals and transfers.
• Signature Verification: Critical operations require explicit user consent through cryptographic signatures, with clear presentation of transaction details.
• Code Security: The wallet undergoes regular security audits, with open-source components allowing community review and validation.
• Gas Estimation: Precise gas estimation prevents failed transactions that could waste user funds through gas costs.
• Spending Limits: Optional spending caps and transaction limits provide additional security for high-value wallets.

These mechanisms work together to provide security while maintaining usability in the self-custody model.