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📞 ℭ𝔞𝔩𝔩 24/7 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱: 1-855-219-0007 𝔄𝔯𝔢 𝔶𝔬𝔲 𝔣𝔞𝔠𝔦𝔫𝔤 𝔦𝔰𝔰𝔲𝔢𝔰 𝔴𝔦𝔱𝔥 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔞𝔫𝔡 𝔫𝔢𝔢𝔡 𝔦𝔪𝔪𝔢𝔡𝔦𝔞𝔱𝔢 𝔞𝔰𝔰𝔦𝔰𝔱𝔞𝔫𝔠𝔢? 𝔇𝔬 𝔫𝔬𝔱 𝔴𝔬𝔯𝔯𝔶--𝔥𝔢𝔩𝔭 𝔦𝔰 𝔧𝔲𝔰𝔱 𝔞 𝔭𝔥𝔬𝔫𝔢 𝔠𝔞𝔩𝔩 𝔞𝔴𝔞𝔶! 𝔒𝔲𝔯 24/7 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱 𝔦𝔰 𝔞𝔳𝔞𝔦𝔩𝔞𝔟𝔩𝔢 𝔞𝔱 1-855-219-0007, 𝔯𝔢𝔞𝔡𝔶 𝔱𝔬 𝔥𝔢𝔩𝔭 𝔶𝔬𝔲 𝔯𝔢𝔰𝔬𝔩𝔳𝔢 𝔞𝔫𝔶 𝔞𝔠𝔠𝔬𝔲𝔫𝔱𝔦𝔫𝔤 𝔬𝔯 𝔱𝔢𝔠𝔥𝔫𝔦𝔠𝔞𝔩 𝔦𝔰𝔰𝔲𝔢𝔰 𝔞𝔯𝔬𝔲𝔫𝔡 𝔱𝔥𝔢 𝔠𝔩𝔬𝔠𝔨. 𝔚𝔥𝔢𝔱𝔥𝔢𝔯 𝔦𝔱 𝔦𝔰 𝔩𝔞𝔱𝔢 𝔞𝔱 𝔫𝔦𝔤𝔥𝔱, 𝔢𝔞𝔯𝔩𝔶 𝔪𝔬𝔯𝔫𝔦𝔫𝔤, 𝔬𝔯 𝔡𝔲𝔯𝔦𝔫𝔤 𝔞 𝔥𝔬𝔩𝔦𝔡𝔞𝔶, 𝔬𝔲𝔯 𝔢𝔵𝔭𝔢𝔯𝔱 𝔱𝔢𝔞𝔪 𝔦𝔰 𝔰𝔱𝔞𝔫𝔡𝔦𝔫𝔤 𝔟𝔶 𝔱𝔬 𝔭𝔯𝔬𝔳𝔦𝔡𝔢 𝔣𝔞𝔰𝔱, 𝔣𝔯𝔦𝔢𝔫𝔡𝔩𝔶 𝔰𝔲𝔭𝔭𝔬𝔯𝔱. 𝔚𝔥𝔶 ℭ𝔞𝔩𝔩 24/7 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱? 📈 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔦𝔰 𝔞 𝔭𝔬𝔴𝔢𝔯𝔣𝔲𝔩 𝔞𝔠𝔠𝔬𝔲𝔫𝔱𝔦𝔫𝔤 𝔰𝔬𝔩𝔲𝔱𝔦𝔬𝔫 𝔲𝔰𝔢𝔡 𝔟𝔶 𝔱𝔥𝔬𝔲𝔰𝔞𝔫𝔡𝔰 𝔬𝔣 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰𝔢𝔰 𝔞𝔠𝔯𝔬𝔰𝔰 𝔱𝔥𝔢 𝔘.𝔖. 𝔅𝔲𝔱 𝔢𝔳𝔢𝔫 𝔱𝔥𝔢 𝔪𝔬𝔰𝔱 𝔯𝔬𝔟𝔲𝔰𝔱 𝔰𝔬𝔣𝔱𝔴𝔞𝔯𝔢 𝔠𝔞𝔫 𝔯𝔲𝔫 𝔦𝔫𝔱𝔬 𝔬𝔠𝔠𝔞𝔰𝔦𝔬𝔫𝔞𝔩 𝔥𝔦𝔠𝔠𝔲𝔭𝔰--𝔞𝔫𝔡 𝔴𝔥𝔢𝔫 𝔦𝔱 𝔡𝔬𝔢𝔰, 𝔶𝔬𝔲 𝔫𝔢𝔢𝔡 𝔥𝔢𝔩𝔭 𝔣𝔞𝔰𝔱. ✅ ℜ𝔢𝔞𝔩-𝔱𝔦𝔪𝔢 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 ✅ ℭ𝔢𝔯𝔱𝔦𝔣𝔦𝔢𝔡 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰 𝔢𝔵𝔭𝔢𝔯𝔱𝔰 ✅ 100% 𝔘.𝔖.-𝔟𝔞𝔰𝔢𝔡 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 ✅ 𝔑𝔬 𝔴𝔞𝔦𝔱𝔦𝔫𝔤--𝔩𝔦𝔳𝔢 𝔞𝔤𝔢𝔫𝔱𝔰 𝔬𝔫 𝔠𝔞𝔩𝔩 𝔍𝔲𝔰𝔱 𝔠𝔞𝔩𝔩 1-855-219-0007 𝔞𝔫𝔶𝔱𝔦𝔪𝔢--𝔬𝔲𝔯 𝔞𝔤𝔢𝔫𝔱𝔰 𝔞𝔯𝔢 𝔱𝔯𝔞𝔦𝔫𝔢𝔡 𝔱𝔬 𝔞𝔰𝔰𝔦𝔰𝔱 𝔴𝔦𝔱𝔥 𝔰𝔢𝔱𝔲𝔭, 𝔦𝔫𝔰𝔱𝔞𝔩𝔩𝔞𝔱𝔦𝔬𝔫, 𝔭𝔞𝔶𝔯𝔬𝔩𝔩, 𝔪𝔲𝔩𝔱𝔦-𝔲𝔰𝔢𝔯 𝔦𝔰𝔰𝔲𝔢𝔰, 𝔞𝔫𝔡 𝔞𝔫𝔶 𝔱𝔢𝔠𝔥𝔫𝔦𝔠𝔞𝔩 𝔢𝔯𝔯𝔬𝔯𝔰 𝔱𝔥𝔞𝔱 𝔪𝔞𝔶 𝔞𝔯𝔦𝔰𝔢. ℭ𝔬𝔪𝔪𝔬𝔫 ℑ𝔰𝔰𝔲𝔢𝔰 𝔚𝔢 ℭ𝔞𝔫 ℌ𝔢𝔩𝔭 𝔚𝔦𝔱𝔥 🔧 𝔚𝔥𝔢𝔫 𝔶𝔬𝔲 𝔞𝔯𝔢 𝔰𝔱𝔲𝔠𝔨, 𝔱𝔦𝔪𝔢 𝔦𝔰 𝔪𝔬𝔫𝔢𝔶. 𝔒𝔲𝔯 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔰𝔭𝔢𝔠𝔦𝔞𝔩𝔦𝔰𝔱𝔰 𝔠𝔞𝔫 𝔴𝔞𝔩𝔨 𝔶𝔬𝔲 𝔱𝔥𝔯𝔬𝔲𝔤𝔥 𝔢𝔳𝔢𝔫 𝔱𝔥𝔢 𝔪𝔬𝔰𝔱 𝔠𝔬𝔪𝔭𝔩𝔢𝔵 𝔭𝔯𝔬𝔟𝔩𝔢𝔪𝔰. ℭ𝔞𝔩𝔩 𝔲𝔰 𝔫𝔬𝔴 𝔞𝔱 1-855-219-0007 𝔣𝔬𝔯 𝔥𝔢𝔩𝔭 𝔴𝔦𝔱𝔥: 🛠️ 𝔈𝔯𝔯𝔬𝔯 𝔠𝔬𝔡𝔢𝔰 𝔩𝔦𝔨𝔢 ℌ202, 6000 𝔰𝔢𝔯𝔦𝔢𝔰, 𝔬𝔯 6177 🧾 ℭ𝔬𝔪𝔭𝔞𝔫𝔶 𝔣𝔦𝔩𝔢 𝔫𝔬𝔱 𝔬𝔭𝔢𝔫𝔦𝔫𝔤 𝔬𝔯 𝔩𝔬𝔞𝔡𝔦𝔫𝔤 🔄 𝔇𝔞𝔱𝔞 𝔰𝔶𝔫𝔠 𝔞𝔫𝔡 𝔟𝔞𝔠𝔨𝔲𝔭 𝔦𝔰𝔰𝔲𝔢𝔰 👥 𝔐𝔲𝔩𝔱𝔦-𝔲𝔰𝔢𝔯 𝔪𝔬𝔡𝔢 𝔠𝔬𝔫𝔣𝔦𝔤𝔲𝔯𝔞𝔱𝔦𝔬𝔫 𝔭𝔯𝔬𝔟𝔩𝔢𝔪𝔰 💰 𝔓𝔞𝔶𝔯𝔬𝔩𝔩 𝔰𝔢𝔱𝔲𝔭 𝔬𝔯 𝔱𝔞𝔵 𝔣𝔬𝔯𝔪 𝔢𝔯𝔯𝔬𝔯𝔰 📊 ℭ𝔲𝔰𝔱𝔬𝔪 𝔯𝔢𝔭𝔬𝔯𝔱𝔰 𝔬𝔯 𝔠𝔥𝔞𝔯𝔱 𝔬𝔣 𝔞𝔠𝔠𝔬𝔲𝔫𝔱𝔰 𝔚𝔥𝔢𝔱𝔥𝔢𝔯 𝔦𝔱 𝔦𝔰 𝔞 𝔰𝔪𝔞𝔩𝔩 𝔟𝔲𝔤 𝔬𝔯 𝔞 𝔪𝔞𝔧𝔬𝔯 𝔠𝔯𝔞𝔰𝔥, 𝔧𝔲𝔰𝔱 𝔠𝔞𝔩𝔩 1-855-219-0007 𝔞𝔫𝔡 𝔤𝔢𝔱 𝔟𝔞𝔠𝔨 𝔱𝔬 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰 𝔮𝔲𝔦𝔠𝔨𝔩𝔶. 𝔄𝔳𝔞𝔦𝔩𝔞𝔟𝔩𝔢 24/7 – 𝔈𝔳𝔢𝔫 𝔬𝔫 𝔚𝔢𝔢𝔨𝔢𝔫𝔡𝔰 𝔞𝔫𝔡 ℌ𝔬𝔩𝔦𝔡𝔞𝔶𝔰 🕒 𝔐𝔞𝔫𝔶 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰 𝔬𝔴𝔫𝔢𝔯𝔰 𝔴𝔬𝔯𝔨 𝔞𝔣𝔱𝔢𝔯 𝔥𝔬𝔲𝔯𝔰 𝔬𝔯 𝔫𝔢𝔢𝔡 𝔥𝔢𝔩𝔭 𝔬𝔲𝔱𝔰𝔦𝔡𝔢 𝔬𝔣 9 𝔱𝔬 5. 𝔗𝔥𝔞𝔱 𝔦𝔰 𝔴𝔥𝔶 𝔬𝔲𝔯 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱 𝔏𝔦𝔫𝔢 (1-855-219-0007) 𝔫𝔢𝔳𝔢𝔯 𝔠𝔩𝔬𝔰𝔢𝔰. 𝔚𝔢 𝔞𝔯𝔢 𝔬𝔭𝔢𝔫 24 𝔥𝔬𝔲𝔯𝔰 𝔞 𝔡𝔞𝔶, 7 𝔡𝔞𝔶𝔰 𝔞 𝔴𝔢𝔢𝔨--𝔦𝔫𝔠𝔩𝔲𝔡𝔦𝔫𝔤 𝔴𝔢𝔢𝔨𝔢𝔫𝔡𝔰 𝔞𝔫𝔡 𝔥𝔬𝔩𝔦𝔡𝔞𝔶𝔰! 🌙 𝔚𝔬𝔯𝔨𝔦𝔫𝔤 𝔩𝔞𝔱𝔢? ℭ𝔞𝔩𝔩 𝔲𝔰. 🎄 ℑ𝔱 𝔦𝔰 𝔞 𝔥𝔬𝔩𝔦𝔡𝔞𝔶? 𝔚𝔢 𝔞𝔯𝔢 𝔰𝔱𝔦𝔩𝔩 𝔥𝔢𝔯𝔢. 🚨 𝔈𝔪𝔢𝔯𝔤𝔢𝔫𝔠𝔶? 𝔚𝔢 𝔴𝔦𝔩𝔩 𝔭𝔦𝔠𝔨 𝔲𝔭. ℑ𝔱 𝔦𝔰 𝔩𝔦𝔨𝔢 𝔥𝔞𝔳𝔦𝔫𝔤 𝔶𝔬𝔲𝔯 𝔬𝔴𝔫 ℑ𝔗 𝔡𝔢𝔭𝔞𝔯𝔱𝔪𝔢𝔫𝔱, 𝔧𝔲𝔰𝔱 𝔞 𝔭𝔥𝔬𝔫𝔢 𝔠𝔞𝔩𝔩 𝔞𝔴𝔞𝔶--1-855-219-0007. 𝔏𝔦𝔳𝔢 ℌ𝔢𝔩𝔭 𝔣𝔯𝔬𝔪 ℭ𝔢𝔯𝔱𝔦𝔣𝔦𝔢𝔡 𝔈𝔵𝔭𝔢𝔯𝔱𝔰 👨💻👩💻 𝔑𝔬𝔱 𝔞𝔩𝔩 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔦𝔰 𝔠𝔯𝔢𝔞𝔱𝔢𝔡 𝔢𝔮𝔲𝔞𝔩. 𝔚𝔥𝔢𝔫 𝔶𝔬𝔲 𝔠𝔞𝔩𝔩 1-855-219-0007, 𝔶𝔬𝔲 𝔞𝔯𝔢 𝔰𝔭𝔢𝔞𝔨𝔦𝔫𝔤 𝔴𝔦𝔱𝔥 𝔭𝔯𝔬𝔣𝔢𝔰𝔰𝔦𝔬𝔫𝔞𝔩𝔰 𝔴𝔥𝔬 𝔞𝔯𝔢: ✔️ ℑ𝔫𝔱𝔲𝔦𝔱 ℭ𝔢𝔯𝔱𝔦𝔣𝔦𝔢𝔡 𝔓𝔯𝔬𝔄𝔡𝔳𝔦𝔰𝔬𝔯𝔰 ✔️ ℌ𝔦𝔤𝔥𝔩𝔶 𝔱𝔯𝔞𝔦𝔫𝔢𝔡 𝔦𝔫 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ ✔️ 𝔈𝔵𝔭𝔢𝔯𝔦𝔢𝔫𝔠𝔢𝔡 𝔦𝔫 𝔥𝔞𝔫𝔡𝔩𝔦𝔫𝔤 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰 𝔞𝔠𝔠𝔬𝔲𝔫𝔱𝔦𝔫𝔤 𝔣𝔬𝔯 𝔞𝔩𝔩 𝔦𝔫𝔡𝔲𝔰𝔱𝔯𝔦𝔢𝔰 ✔️ 𝔉𝔯𝔦𝔢𝔫𝔡𝔩𝔶, 𝔭𝔞𝔱𝔦𝔢𝔫𝔱, 𝔞𝔫𝔡 𝔠𝔩𝔢𝔞𝔯 𝔠𝔬𝔪𝔪𝔲𝔫𝔦𝔠𝔞𝔱𝔬𝔯𝔰 𝔚𝔢 𝔡𝔬 𝔫𝔬𝔱 𝔯𝔢𝔞𝔡 𝔣𝔯𝔬𝔪 𝔰𝔠𝔯𝔦𝔭𝔱𝔰. 𝔚𝔢 𝔰𝔬𝔩𝔳𝔢 𝔯𝔢𝔞𝔩 𝔭𝔯𝔬𝔟𝔩𝔢𝔪𝔰 𝔴𝔦𝔱𝔥 𝔯𝔢𝔞𝔩 𝔭𝔢𝔬𝔭𝔩𝔢--𝔣𝔞𝔰𝔱. ℜ𝔢𝔪𝔬𝔱𝔢 𝔄𝔰𝔰𝔦𝔰𝔱𝔞𝔫𝔠𝔢 𝔄𝔳𝔞𝔦𝔩𝔞𝔟𝔩𝔢 🔐 𝔖𝔬𝔪𝔢 𝔭𝔯𝔬𝔟𝔩𝔢𝔪𝔰 𝔫𝔢𝔢𝔡 𝔥𝔞𝔫𝔡𝔰-𝔬𝔫 𝔱𝔯𝔬𝔲𝔟𝔩𝔢𝔰𝔥𝔬𝔬𝔱𝔦𝔫𝔤. 𝔚𝔢 𝔬𝔣𝔣𝔢𝔯 𝔰𝔢𝔠𝔲𝔯𝔢 𝔯𝔢𝔪𝔬𝔱𝔢 𝔰𝔲𝔭𝔭𝔬𝔯𝔱--𝔴𝔦𝔱𝔥 𝔶𝔬𝔲𝔯 𝔭𝔢𝔯𝔪𝔦𝔰𝔰𝔦𝔬𝔫--𝔰𝔬 𝔬𝔲𝔯 𝔢𝔵𝔭𝔢𝔯𝔱𝔰 𝔠𝔞𝔫 𝔞𝔠𝔠𝔢𝔰𝔰 𝔶𝔬𝔲𝔯 𝔰𝔶𝔰𝔱𝔢𝔪 𝔞𝔫𝔡 𝔣𝔦𝔵 𝔱𝔥𝔢 𝔦𝔰𝔰𝔲𝔢 𝔦𝔫 𝔯𝔢𝔞𝔩 𝔱𝔦𝔪𝔢. 𝔄𝔩𝔩 𝔶𝔬𝔲 𝔫𝔢𝔢𝔡 𝔱𝔬 𝔡𝔬 𝔦𝔰 𝔠𝔞𝔩𝔩 1-855-219-0007, 𝔞𝔫𝔡 𝔴𝔢 𝔴𝔦𝔩𝔩 𝔤𝔲𝔦𝔡𝔢 𝔶𝔬𝔲 𝔰𝔱𝔢𝔭-𝔟𝔶-𝔰𝔱𝔢𝔭. 𝔜𝔬𝔲𝔯 𝔡𝔞𝔱𝔞 𝔦𝔰 𝔰𝔞𝔣𝔢 𝔞𝔫𝔡 𝔠𝔬𝔫𝔣𝔦𝔡𝔢𝔫𝔱𝔦𝔞𝔩--𝔞𝔩𝔴𝔞𝔶𝔰 𝔭𝔯𝔬𝔱𝔢𝔠𝔱𝔢𝔡 𝔲𝔫𝔡𝔢𝔯 𝔰𝔱𝔯𝔦𝔠𝔱 𝔰𝔢𝔠𝔲𝔯𝔦𝔱𝔶 𝔭𝔯𝔬𝔱𝔬𝔠𝔬𝔩𝔰. 🔒 𝔊𝔢𝔱 ℌ𝔢𝔩𝔭 𝔑𝔬𝔴 – 𝔇𝔬 𝔫𝔬𝔱 𝔚𝔞𝔦𝔱! 🚀 𝔚𝔥𝔢𝔫 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔰𝔱𝔬𝔭𝔰 𝔴𝔬𝔯𝔨𝔦𝔫𝔤, 𝔶𝔬𝔲𝔯 𝔴𝔬𝔯𝔨𝔣𝔩𝔬𝔴 𝔰𝔱𝔞𝔩𝔩𝔰. 𝔗𝔥𝔞𝔱 𝔦𝔰 𝔴𝔥𝔶 𝔶𝔬𝔲 𝔰𝔥𝔬𝔲𝔩𝔡 𝔫𝔬𝔱 𝔴𝔞𝔦𝔱 𝔥𝔬𝔲𝔯𝔰 𝔬𝔯 𝔰𝔢𝔞𝔯𝔠𝔥 𝔣𝔬𝔯𝔲𝔪𝔰 𝔣𝔬𝔯 𝔞𝔫𝔰𝔴𝔢𝔯𝔰. 𝔊𝔢𝔱 𝔡𝔦𝔯𝔢𝔠𝔱 𝔞𝔠𝔠𝔢𝔰𝔰 𝔱𝔬 𝔬𝔲𝔯 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔱𝔢𝔞𝔪 𝔫𝔬𝔴 𝔟𝔶 𝔠𝔞𝔩𝔩𝔦𝔫𝔤: 📞 1-855-219-0007 𝔇𝔬 𝔫𝔬𝔱 𝔩𝔢𝔱 𝔱𝔢𝔠𝔥𝔫𝔦𝔠𝔞𝔩 𝔦𝔰𝔰𝔲𝔢𝔰 𝔰𝔩𝔬𝔴 𝔶𝔬𝔲 𝔡𝔬𝔴𝔫. 𝔒𝔲𝔯 𝔣𝔯𝔦𝔢𝔫𝔡𝔩𝔶 𝔞𝔤𝔢𝔫𝔱𝔰 𝔞𝔯𝔢 𝔞𝔳𝔞𝔦𝔩𝔞𝔟𝔩𝔢 𝔡𝔞𝔶 𝔬𝔯 𝔫𝔦𝔤𝔥𝔱 𝔱𝔬 𝔤𝔢𝔱 𝔶𝔬𝔲 𝔟𝔞𝔠𝔨 𝔬𝔫 𝔱𝔯𝔞𝔠𝔨--𝔲𝔰𝔲𝔞𝔩𝔩𝔶 𝔴𝔦𝔱𝔥𝔦𝔫 𝔪𝔦𝔫𝔲𝔱𝔢𝔰. 𝔚𝔥𝔬 ℭ𝔞𝔫 ℭ𝔞𝔩𝔩 𝔗𝔥𝔦𝔰 𝔖𝔲𝔭𝔭𝔬𝔯𝔱 𝔏𝔦𝔫𝔢? 📣 𝔗𝔥𝔦𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔰𝔢𝔯𝔳𝔦𝔠𝔢 𝔦𝔰 𝔭𝔢𝔯𝔣𝔢𝔠𝔱 𝔣𝔬𝔯: ✅ 𝔖𝔪𝔞𝔩𝔩 𝔱𝔬 𝔪𝔦𝔡-𝔰𝔦𝔷𝔢 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰𝔢𝔰 ✅ 𝔄𝔠𝔠𝔬𝔲𝔫𝔱𝔞𝔫𝔱𝔰 𝔞𝔫𝔡 𝔟𝔬𝔬𝔨𝔨𝔢𝔢𝔭𝔢𝔯𝔰 ✅ 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢-𝔩𝔢𝔳𝔢𝔩 𝔬𝔯𝔤𝔞𝔫𝔦𝔷𝔞𝔱𝔦𝔬𝔫𝔰 ✅ 𝔉𝔦𝔯𝔰𝔱-𝔱𝔦𝔪𝔢 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰 𝔲𝔰𝔢𝔯𝔰 ✅ 𝔏𝔬𝔫𝔤-𝔱𝔦𝔪𝔢 𝔲𝔰𝔢𝔯𝔰 𝔴𝔦𝔱𝔥 𝔫𝔢𝔴 𝔢𝔯𝔯𝔬𝔯𝔰 ℑ𝔣 𝔶𝔬𝔲 𝔞𝔯𝔢 𝔲𝔰𝔦𝔫𝔤 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™, 𝔱𝔥𝔦𝔰 𝔫𝔲𝔪𝔟𝔢𝔯 𝔦𝔰 𝔣𝔬𝔯 𝔶𝔬𝔲: 1-855-219-0007. 𝔇𝔬 𝔫𝔬𝔱 ℜ𝔦𝔰𝔨 𝔏𝔬𝔰𝔦𝔫𝔤 𝔇𝔞𝔱𝔞 𝔬𝔯 𝔗𝔦𝔪𝔢 – ℭ𝔞𝔩𝔩 𝔑𝔬𝔴! ⏳ 𝔈𝔳𝔢𝔫 𝔪𝔦𝔫𝔬𝔯 𝔦𝔰𝔰𝔲𝔢𝔰 𝔦𝔫 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰 𝔠𝔞𝔫 𝔩𝔢𝔞𝔡 𝔱𝔬 𝔥𝔬𝔲𝔯𝔰 𝔬𝔣 𝔩𝔬𝔰𝔱 𝔱𝔦𝔪𝔢 𝔬𝔯 𝔦𝔫𝔠𝔬𝔯𝔯𝔢𝔠𝔱 𝔣𝔦𝔫𝔞𝔫𝔠𝔦𝔞𝔩𝔰. 𝔚𝔥𝔢𝔱𝔥𝔢𝔯 𝔶𝔬𝔲 𝔞𝔯𝔢 𝔢𝔵𝔭𝔢𝔯𝔦𝔢𝔫𝔠𝔦𝔫𝔤 𝔭𝔢𝔯𝔣𝔬𝔯𝔪𝔞𝔫𝔠𝔢 𝔩𝔞𝔤𝔰, 𝔠𝔯𝔞𝔰𝔥𝔦𝔫𝔤 𝔰𝔬𝔣𝔱𝔴𝔞𝔯𝔢, 𝔬𝔯 𝔲𝔰𝔢𝔯 𝔞𝔠𝔠𝔢𝔰𝔰 𝔦𝔰𝔰𝔲𝔢𝔰, 𝔡𝔬 𝔫𝔬𝔱 𝔱𝔯𝔶 𝔱𝔬 𝔣𝔦𝔵 𝔦𝔱 𝔞𝔩𝔬𝔫𝔢. ℑ𝔫𝔰𝔱𝔢𝔞𝔡, 𝔠𝔞𝔩𝔩 𝔱𝔥𝔢 𝔢𝔵𝔭𝔢𝔯𝔱𝔰 𝔞𝔱 1-855-219-0007--𝔶𝔬𝔲𝔯 𝔱𝔯𝔲𝔰𝔱𝔢𝔡 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱 𝔡𝔦𝔯𝔢𝔠𝔱𝔬𝔯𝔶. 𝔚𝔢 𝔴𝔦𝔩𝔩 𝔡𝔦𝔞𝔤𝔫𝔬𝔰𝔢 𝔞𝔫𝔡 𝔯𝔢𝔰𝔬𝔩𝔳𝔢 𝔱𝔥𝔢 𝔭𝔯𝔬𝔟𝔩𝔢𝔪, 𝔬𝔣𝔱𝔢𝔫 𝔬𝔫 𝔱𝔥𝔢 𝔣𝔦𝔯𝔰𝔱 𝔠𝔞𝔩𝔩. 📞 ℭ𝔞𝔩𝔩 𝔑𝔬𝔴: 1-855-219-0007 – 𝔔𝔲𝔦𝔠𝔨𝔅𝔬𝔬𝔨𝔰® 𝔈𝔫𝔱𝔢𝔯𝔭𝔯𝔦𝔰𝔢™ 𝔖𝔲𝔭𝔭𝔬𝔯𝔱 𝔜𝔬𝔲 ℭ𝔞𝔫 𝔗𝔯𝔲𝔰𝔱! 𝔏𝔢𝔱 𝔲𝔰 𝔥𝔞𝔫𝔡𝔩𝔢 𝔱𝔥𝔢 𝔰𝔱𝔯𝔢𝔰𝔰 𝔴𝔥𝔦𝔩𝔢 𝔶𝔬𝔲 𝔣𝔬𝔠𝔲𝔰 𝔬𝔫 𝔤𝔯𝔬𝔴𝔦𝔫𝔤 𝔶𝔬𝔲𝔯 𝔟𝔲𝔰𝔦𝔫𝔢𝔰𝔰. 𝔉𝔞𝔰𝔱. 𝔉𝔯𝔦𝔢𝔫𝔡𝔩𝔶. 24/7. • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔥𝔢𝔩𝔭𝔩𝔦𝔫𝔢 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔠𝔬𝔫𝔱𝔞𝔠𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔲𝔰𝔱𝔬𝔪𝔢𝔯 𝔠𝔞𝔯𝔢 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔭𝔥𝔬𝔫𝔢 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔢𝔪𝔞𝔦𝔩 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔩𝔦𝔳𝔢 𝔠𝔥𝔞𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔥𝔢𝔩𝔭 𝔡𝔢𝔰𝔨 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔬𝔫𝔱𝔞𝔠𝔱 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔱𝔢𝔞𝔪 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔲𝔰𝔱𝔬𝔪𝔢𝔯 𝔞𝔰𝔰𝔦𝔰𝔱𝔞𝔫𝔠𝔢 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔢𝔯𝔳𝔦𝔠𝔢 𝔥𝔬𝔱𝔩𝔦𝔫𝔢 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔱𝔢𝔠𝔥𝔫𝔦𝔠𝔞𝔩 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔦𝔰𝔰𝔲𝔢 𝔯𝔢𝔰𝔬𝔩𝔲𝔱𝔦𝔬𝔫 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔞𝔠𝔠𝔬𝔲𝔫𝔱 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔭𝔞𝔶𝔪𝔢𝔫𝔱 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔴𝔦𝔱𝔥𝔡𝔯𝔞𝔴𝔞𝔩 𝔦𝔰𝔰𝔲𝔢𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔩𝔬𝔤𝔦𝔫 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔳𝔢𝔯𝔦𝔣𝔦𝔠𝔞𝔱𝔦𝔬𝔫 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔞𝔠𝔠𝔬𝔲𝔫𝔱 𝔯𝔢𝔠𝔬𝔳𝔢𝔯𝔶 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔩𝔬𝔰𝔱 𝔞𝔠𝔠𝔢𝔰𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔢𝔠𝔲𝔯𝔦𝔱𝔶 𝔦𝔰𝔰𝔲𝔢𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔣𝔯𝔞𝔲𝔡 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔟𝔦𝔩𝔩𝔦𝔫𝔤 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔱𝔯𝔞𝔫𝔰𝔞𝔠𝔱𝔦𝔬𝔫 𝔭𝔯𝔬𝔟𝔩𝔢𝔪𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔴𝔞𝔩𝔩𝔢𝔱 𝔦𝔰𝔰𝔲𝔢𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔪𝔬𝔟𝔦𝔩𝔢 𝔞𝔭𝔭 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔯𝔶𝔭𝔱𝔬 𝔱𝔯𝔞𝔫𝔰𝔣𝔢𝔯 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔢𝔯𝔯𝔬𝔯 𝔣𝔦𝔵𝔦𝔫𝔤 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔩𝔬𝔤𝔦𝔫 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔭𝔯𝔬𝔟𝔩𝔢𝔪 𝔯𝔢𝔰𝔬𝔩𝔲𝔱𝔦𝔬𝔫 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔣𝔢𝔢𝔡𝔟𝔞𝔠𝔨 𝔠𝔬𝔫𝔱𝔞𝔠𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔞𝔠𝔠𝔬𝔲𝔫𝔱 𝔰𝔲𝔰𝔭𝔢𝔫𝔰𝔦𝔬𝔫 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔴𝔦𝔱𝔥𝔡𝔯𝔞𝔴𝔞𝔩 𝔡𝔢𝔩𝔞𝔶 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔡𝔢𝔭𝔬𝔰𝔦𝔱 𝔦𝔰𝔰𝔲𝔢𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔯𝔶𝔭𝔱𝔬 𝔠𝔬𝔫𝔳𝔢𝔯𝔰𝔦𝔬𝔫 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔬𝔯𝔡𝔢𝔯 𝔰𝔱𝔞𝔱𝔲𝔰 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔯𝔢𝔣𝔲𝔫𝔡 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔱𝔞𝔵 𝔡𝔬𝔠𝔲𝔪𝔢𝔫𝔱 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔦𝔫𝔱𝔢𝔯𝔫𝔞𝔱𝔦𝔬𝔫𝔞𝔩 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 24/7 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔞𝔠𝔠𝔬𝔲𝔫𝔱 𝔰𝔢𝔱𝔱𝔦𝔫𝔤𝔰 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔭𝔞𝔰𝔰𝔴𝔬𝔯𝔡 𝔯𝔢𝔰𝔢𝔱 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔱𝔴𝔬-𝔣𝔞𝔠𝔱𝔬𝔯 𝔞𝔲𝔱𝔥𝔢𝔫𝔱𝔦𝔠𝔞𝔱𝔦𝔬𝔫 𝔥𝔢𝔩𝔭 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔠𝔲𝔰𝔱𝔬𝔪𝔢𝔯 𝔦𝔫𝔮𝔲𝔦𝔯𝔶 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔰𝔲𝔭𝔭𝔬𝔯𝔱 𝔫𝔲𝔪𝔟𝔢𝔯 𝔘𝔖𝔄 • 𝔔𝔲!𝔠𝔨𝔅00𝔨𝔰 𝔥𝔢𝔩𝔭 𝔠𝔢𝔫𝔱𝔢𝔯 𝔭𝔥𝔬𝔫𝔢 𝔫𝔲𝔪𝔟𝔢𝔯
Excited to be here
Get ready to roll up your sleeves at MATLAB EXPO 2025 – our global online event is back, and this year we’re offering 10 hands-on workshops designed to spark innovation and deepen your skills with MATLAB Online and Simulink Online.
Whether you're exploring AI, modeling batteries, or building carbon trackers, these live workshops are your chance to:
- Work directly in MATLAB and Simulink Online
- Solve real-world challenges with guidance from MathWorks experts
- Connect with peers across industries
- Ask questions and get live feedback
Join the Experience to learn more about each workshop below!
Which workshop are you most excited to attend?!
Day 1:
- Beyond the Labels: Leveraging AI Techniques for Enlightened Product Choices
- A Hands-On Introduction to Reinforcement Learning with MATLAB and Simulink
- Curriculum Development with MATLAB Copilot and Generative AI
- Simscape Battery Workshop
- Generating Tests for your MATLAB code
Day 2:
- Hands-On AI for Smart Appliances: From Sensor Data to Embedded Code
- A Hands-On Introduction to Reduced Order Modeling with MATLAB and Simulink
- Introduction to Research Software and Development with Simulink
- Hack Your Carbon Impact: Build and Publish an Emissions Tracker with MATLAB
- How to Simulate Scalable Cellular and Connectivity Networks: A Hands-On Session
We look forward to Accelerating the Pace of Engineering and Science together!
I am excited to join this community to learn the more particularly the Matlab/Simulink
It’s an honor to deliver the keynote at MATLAB EXPO 2025. I'll explore how AI changes the game in engineered systems, bringing intelligence to every step of the process from design to deployment. This short video captures a glimpse of what I’ll share:
What excites or challenges you about this shift? Drop a comment or start a thread!
I just learned you can access MATLAB Online from the following shortcut in your web browser: https://matlab.new
Thanks @Yann Debray
From his recent blog post: pip & uv in MATLAB Online » Artificial Intelligence - MATLAB & Simulink
I'm developing a comprehensive MATLAB programming course and seeking passionate co-trainers to collaborate!
Why MATLAB Matters:Many people underestimate MATLAB's significance in:
- Communication systems
- Signal processing
- Mathematical modeling
- Engineering applications
- Scientific computing
Course Structure:
- Foundation Module: MATLAB basics and fundamentals
- Image Processing: Practical applications and techniques
- Signal Processing: Analysis and implementation
- Machine Learning: ML algorithms using MATLAB
- Hands-on Learning: Projects, assignments.
What I'm Looking For:
- Enthusiastic educators willing to share knowledge
- Experience in any MATLAB application area
- Commitment to collaborative teaching
Interested in joining as a co-trainer? Please comment below or reach out directly!
Online Doc + System Browser
15%
Online Doc + Dedicated Browser
15%
Offline Doc +System Browser
5%
Offline Doc + Dedicated Browser
20%
Hybrid Approach (Support All Modes)
25%
User-Definable / Fully Configurable
20%
20 votes
Please share with us how you are using AI in your control design workflows and what you want to hear most in our upcoming talk, 4 Ways to Improve Control Design Workflows with AI.
Arkadiy
Hello Everyone, I’m Vikram Kumar Singh, and I’m excited to be part of this amazing MATLAB community!
I’m deeply interested in learning more from all of you and contributing wherever I can. Recently, I completed a project on modeling and simulation of a Li-ion battery with a Battery Management System (BMS) for fault detection and management.
I’d love to share my learnings and also explore new ideas together with this group. Looking forward to connecting and growing with the community!
Excited for MATLAB EXPO 2025!
I’m a Master’s student in Electrical Engineering at UNSW Sydney, researching EV fleet charging and hybrid energy strategies integrating battery-electric and hydrogen fuel cell vehicles.
LinkedIn link: www.linkedin.com/in/yuanzhe-chen-6b2158351
ResearchGate link: https://www.researchgate.net/profile/Yuanzhe-Chen-9?ev=hdr_xprf
#MATLABEXPO #EV #FCEV #SmartGrid
I recently published this blog post about resources to help people learn MATLAB https://blogs.mathworks.com/matlab/2025/09/11/learning-matlab-in-2025/
What are your favourite MATLAB learning resources?
Excited to link and sync to be a part of better learning experience
happy to be here
Excited to link up
What if you had no isprime utility to rely on in MATLAB? How would you identify a number as prime? An easy answer might be something tricky, like that in simpleIsPrime0.
simpleIsPrime0 = @(N) ismember(N,primes(N));
But I’ll also disallow the use of primes here, as it does not really test to see if a number is prime. As well, it would seem horribly inefficient, generating a possibly huge list of primes, merely to learn something about the last member of the list.
Looking for a more serious test for primality, I’ve already shown how to lighten the load by a bit using roughness, to sometimes identify numbers as composite and therefore not prime.
https://www.mathworks.com/matlabcentral/discussions/tips/879745-primes-and-rough-numbers-basic-ideas
But to actually learn if some number is prime, we must do a little more. Yes, this is a common homework problem assigned to students, something we have seen many times on Answers. It can be approached in many ways too, so it is worth looking at the problem in some depth.
The definition of a prime number is a natural number greater than 1, which has only two factors, thus 1 and itself. That makes a simple test for primality of the number N easy. We just try dividing the number by every integer greater than 1, and not exceeding N-1. If any of those trial divides leaves a zero remainder, then N cannot be prime. And of course we can use mod or rem instead of an explicit divide, so we need not worry about floating point trash, as long as the numbers being tested are not too large.
simpleIsPrime1 = @(N) all(mod(N,2:N-1) ~= 0);
Of course, simpleIsPrime1 is not a good code, in the sense that it fails to check if N is an integer, or if N is less than or equal to 1. It is not vectorized, and it has no documentation at all. But it does the job well enough for one simple line of code. There is some virtue in simplicity after all, and it is certainly easy to read. But sometimes, I wish a function handle could include some help comments too! A feature request might be in the offing.
simpleIsPrime1(9931)
simpleIsPrime1(9932)
simpleIsPrime1 works quite nicely, and seems pretty fast. What could be wrong? At some point, the student is given a more difficult problem, to identify if a significantly larger integer is prime. simpleIsPrime1 will then cause a computer to grind to a distressing halt if given a sufficiently large number to test. Or it might even error out, when too large a vector of numbers was generated to test against. For example, I don't think you want to test a number of the order of 2^64 using simpleIsPrime1, as performing on the order of 2^64 divides will be highly time consuming.
uint64(2)^63-25
Is it prime? I’ve not tested it to learn if it is, and simpleIsPrime1 is not the tool to perform that test anyway.
A student might realize the largest possible integer factors of some number N are the numbers N/2 and N itself. But, if N/2 is a factor, then so is 2, and some thought would suggest it is sufficient to test only for factors that do not exceed sqrt(N). This is because if a is a divisor of N, then so is b=N/a. If one of them is larger than sqrt(N), then the other must be smaller. That could lead us to an improved scheme in simpleIsPrime2.
simpleIsPrime2 = @(N) all(mod(N,2:sqrt(N)));
For an integer of the size 2^64, now you only need to perform roughly 2^32 trial divides. Maybe we might consider the subtle improvement found in simpleIsPrime3, which avoids trial divides by the even integers greater than 2.
simpleIsPrime3 = @(N) (N == 2) || (mod(N,2) && all(mod(N,3:2:sqrt(N))));
simpleIsPrime3 needs only an approximate maximum of 2^31 trial divides even for numbers as large as uint64 can represent. While that is large, it is still generally doable on the computers we have today, even if it might be slow.
Sadly, my goals are higher than even the rather lofty limit given by UINT64 numbers. The problem of course is that a trial divide scheme, despite being 100% accurate in its assessment of primality, is a time hog. Even an O(sqrt(N)) scheme is far too slow for numbers with thousands or millions of digits. And even for a number as “small” as 1e100, a direct set of trial divides by all primes less than sqrt(1e100) would still be practically impossible, as there are roughly n/log(n) primes that do not exceed n. For an integer on the order of 1e50,
1e50/log(1e50)
It is practically impossible to perform that many divides on any computer we can make today. Can we do better? Is there some more efficient test for primality? For example, we could write a simple sieve of Eratosthenes to check each prime found not exceeding sqrt(N).
function [TF,SmallPrime] = simpleIsPrime4(N)
% simpleIsPrime3 - Sieve of Eratosthenes to identify if N is prime
% [TF,SmallPrime] = simpleIsPrime3(N)
%
% Returns true if N is prime, as well as the smallest prime factor
% of N when N is composite. If N is prime, then SmallPrime will be N.
Nroot = ceil(sqrt(N)); % ceil caters for floating point issues with the sqrt
TF = true;
SieveList = true(1,Nroot+1); SieveList(1) = false;
SmallPrime = 2;
while TF
% Find the "next" true element in SieveList
while (SmallPrime <= Nroot+1) && ~SieveList(SmallPrime)
SmallPrime = SmallPrime + 1;
end
% When we drop out of this loop, we have found the next
% small prime to check to see if it divides N, OR, we
% have gone past sqrt(N)
if SmallPrime > Nroot
% this is the case where we have now looked at all
% primes not exceeding sqrt(N), and have found none
% that divide N. This is where we will drop out to
% identify N as prime. TF is already true, so we need
% not set TF.
SmallPrime = N;
return
else
if mod(N,SmallPrime) == 0
% smallPrime does divide N, so we are done
TF = false;
return
end
% update SieveList
SieveList(SmallPrime:SmallPrime:Nroot) = false;
end
end
end
simpleIsPrime4 does indeed work reasonably well, though it is sometimes a little slower than is simpleIsPrime3, and everything is hugely faster than simpleIsPrime1.
timeit(@() simpleIsPrime1(111111111))
timeit(@() simpleIsPrime2(111111111))
timeit(@() simpleIsPrime3(111111111))
timeit(@() simpleIsPrime4(111111111))
All of those times will slow to a crawl for much larger numbers of course. And while I might find a way to subtly improve upon these codes, any improvement will be marginal in the end if I try to use any such direct approach to primality. We must look in a different direction completely to find serious gains.
At this point, I want to distinguish between two distinct classes of tests for primality of some large number. One class of test is what I might call an absolute or infallible test, one that is perfectly reliable. These are tests where if X is identified as prime/composite then we can trust the result absolutely. The tests I showed in the form of simpleIsPrime1, simpleIsPrime2, simpleIsPrime3 and aimpleIsprime4, were all 100% accurate, thus they fall into the class of infallible tests.
The second general class of test for primality is what I will call an evidentiary test. Such a test provides evidence, possibly quite strong evidence, that the given number is prime, but in some cases, it might be mistaken. I've already offered a basic example of a weak evidentiary test for primality in the form of roughness. All primes are maximally rough. And therefore, if you can identify X as being rough to some extent, this provides evidence that X is also prime, and the depth of the roughness test influences the strength of the evidence for primality. While this is generally a fairly weak test, it is a test nevertheless, and a good exclusionary test, a good way to avoid more sophisticated but time consuming tests.
These evidentiary tests all have the property that if they do identify X as being composite, then they are always correct. In the context of roughness, if X is not sufficiently rough, then X is also not prime. On the other side of the coin, if you can show X is at least (sqrt(X)+1)-rough, then it is positively prime. (I say this to suggest that some evidentiary tests for primality can be turned into truth telling tests, but that may take more effort than you can afford.) The problem is of course that is literally impossible to verify that degree of roughness for numbers with many thousands of digits.
In my next post, I'll look at the Fermat test for primality, based on Fermat's little theorem.
Share your learning starting trouble experience of Matlab.. Looking forward for more answers..
Hello everyone , i am excited to learn more!
Helllo all
I write The MATLAB Blog and have covered various enhancements to MATLAB's ODE capabilities over the last couple of years. Here are a few such posts
- The new solution framework for Ordinary Differential Equations (ODEs) in MATLAB R2023b
- Faster Ordinary Differential Equations (ODEs) solvers and Sensitivity Analysis of Parameters: Introducing SUNDIALS support in MATLAB
- Solving Higher-Order ODEs in MATLAB
- Function handles are faster in MATLAB R2023a (Faster function handles led to faster ode45 and friends)
- Understanding Tolerances in Ordinary Differential Equation Solvers
Everyone in this community has deeply engaged with all of these posts and given me lots of ideas for future enhancements which I've dutifully added to our internal enhancment request database.
Because I've asked for so much in this area, I was recently asked if there's anything else we should consider in the area of ODEs. Since all my best ideas come from all of you, I'm asking here....
So. If you could ask for new and improved functionality for solving ODEs with MATLAB, what would it be and (ideally) why?
Cheers,
Mike
