| Number_Theory Proofs along with equation solvers and graphical views written in Java. |
| Number_Theory Dave Rusin's guide to number theory. |
| Number_Theory_Foundation Aims to collect donations from supporters of number theory and to disburse these donations to encourage research. |
| Number_Theory_Glossary Compiled by Robert Campbell. |
| Online_Seminar__Galois_theory_of_p-Extensions Email seminar on "Galoissche Theorie der p-Erweiterungen" by Helmut Koch. |
| Other_Maths A wide selection of number theory topics compiled by Jon Perry. |
| Some_Highlights_of_Arithmetic_Combinatorics Lecture notes and resources on combinatorial number theory by Terence Tao. |
| Some_Number-Theoretical_Constants Products of rational functions of p over primes, computed by Gerhard Niklasch and Pieter Moree. |
| Somos_Polynomials Related to Somos sequences and elliptic theta functions. |
| The_Somos_Sequence_Site Web resources for information on Somos sequences and related topics such as elliptic divisibility sequences. |
| Square-free_Gaps Algorithm and source code for the calculation of square-free numbers and gaps. |
| Transcendental_Numbers Maple worksheets, lecture notes and links to other resources by John Cosgrove. |
| The_Valuation_Theory_Home_Page A forum for all mathematicians who work in valuation theory or apply valuation theoretical results in their own field of research. |
| Vignettes_on_Automorphic_and_Modular_forms,_Representations,_L-functions,_and_Number_Theory By Paul Garrett. |
| Visible_Structures_in_Number_Theory By Peter Borwein and Loki Jörgenson. Recognising number patterns visually. |
| The_Work_of_Robert_Langlands Thesis, papers, manuscripts, letters and bibliography. |
| World_Records_for_Numerical_Palindromes The 196 Palindrome Quest and The Most Delayed Palindromic Number, by Jason Doucette. |
| American_Association_of_Tissue_Banks Professional organization for accreditation and information. |
| Asymptote_Cool_Guide_to_Cryopreservation Manual of the general principles of cryobiology, especially regarding the preservation of sperm and embryos. Detailed but accessible to the educated layman. |
| BioLife_Solutions,_Inc_ Conducts research and provides hypothermic and cryopreservation solutions for cell and tissue preservation, including the HypoThermosol and CryoStor products. |
| Biothermal_Sciences_Laboratory,_U__of_Illinois_at_Chicago Research on cryobiology, heat shock, and fever. |
| Conservation_of_endangered_animals_by_using_sperm_and_embryo_freezing A project to preserve several endangered mammals of far northern Europe, using cryopreservation. At the University of Kuopio, Finland. |
| Cryobiology International journal of low temperature biology and medicine. Published by the Society for Cryobiology. |
| Cryobiology_and_Anhydrobiology_-_School_of_Physics,University_of_New_South_Wales,_Sydney_Australia Research and introduction by Associate professor Joe Wolfe |
| Cryobiology__Low_temperature_studies_of_biological_systems A readable summary of the field of cryobiology by a college biology student, but with some confused opinions about the intent of the field. |
| Cryolab_-_Roskilde_University Cryobiology laboratory at Roskilde University, Denmark. Concentration on cold adaptation in animals, with ongoing research in antifreeze proteins, microtubules, cryopreservation. |
| CryoLife,_Inc_ Manufacturer of medical tissues, heart valves, and oncology drugs. Shown are tissue services, research and company profile. |
| Cryopreservation_Application_Guide A detailed introduction to cell preservation technology from Nalgene Nunc. PDF file (249K) |
| Cryopreservation_Research Research on trehalose and cryopreservation at University of Wisconsin, Molecular Thermodynamics and Statistical Mechanics Group. Click on "Research." Includes a link to their nanolithography program |
| Cryosite A company which provides secure off-site, ultra-low temperature and cryogenic storage for biological specimens. In Sydney, Australia. |
| Frozen_Fruit_Fly_Embryos_Hatched An article from "Technical Highlights" in the Oak Ridge National Laboratory Review (Vol. 26, No. 1). Reports the research of Peter Mazur and others from 1992. |
| Genternity_LLC Proposed services to include DNA preserved before cremation funeral for genetic preservation and immortality. |
| International_Bioresearch_Solutions IBS is a non-profit national organization that works with organ procurement organizations to place organs that are not suitable for transplant with medical researchers. |
| INVAM_-_Freezing_germplasm_in_liquid_nitrogen Research suggestions from the INVAM Culture Collection at West Virginia University. |
| Ken_Storey\'s_Lab The "frozen frog" expert at Carleton University, Ottawa, Ontario, Canada. |
| Lyophilization_Services_of_New_England,_Inc_ Contract manufacturer providing lyophilization (freeze-drying) of products, both vial and bulk. In Manchester, New Hampshire. |
| National_Seed_Storage_Laboratory_(NSSL) Government facility provides cold storage for seeds, graftable buds and animal semen to support diversity in farm flora and fauna. Page includes virtual tour of the facility with an explanation of th |
| Phase_Technologies An organization dedicated to the lyophilization (freeze-drying) process that is used in the manufacture of pharmaceutical, biotechnology, and diagnostic products. Also provides lyophilization service |
| Society_for_Cryobiology The primary scientific society in this field, founded in 1964 to promote research into the effects of low temperatures on all types of organisms and their constituent cells, tissues and organs. |
| 21st_Century_Medicine A research and development company in the fields of cryopreservation and ice control technologies for cryobiology and industrial uses. In Rancho Cucamonga, California. |
"The following problem arises immediately.If ε is a positive quantity, to find a polynomial of the formf(x) = xr +a1 xr-1 +... + arwhere the a's are integers, such that the absolute value of theproduct of those roots of f which lie outside the unit circle, liesbetween 1 and 1 + ε."This problem, in interest in itself, is especially important for ourpurposes.Whether or not the problem has a solution for ε < 0.176 we do notknow." -- Derrick Henry Lehmer, 1933.Mahler's measure of a polynomial f is defined to be theabsolute value of the product of those roots of f which lie outsidethe unit disk, multiplied by the absolute value of the coefficient of theleading term of f.We denote it M(f).Lehmer's problem, sometimes called Lehmer's question, or Lehmer'sconjecture, asks if there exists a constant C > 1 such that everypolynomial f with integer coefficients and M(f) > 1has M(f) 
C.Lehmer added the following remark in his 1933 paper (using Ω todenote the measure):"We have not made an examination of all 10th degree symmetric polynomials,but a rather intensive search has failed to reveal a better polynomial thanx10 + x9 - x7 - x6 - x5 - x4 - x3 + x + 1, Ω = 1.176280818."All efforts to find a better equation of degree 12 and 14 have beenunsuccessful."Despite extensive searches, Lehmer's polynomial remains the world champion.This page summarizes what is known today about Lehmer's problem.It includes descriptions of algorithms, histories of searches performed,and various lists of polynomials with small measure.
This page was ranked as a top 50 resource in number theory by the (nowdefunct) search engine Links2Go.Please write if you have comments or contributions!Michael MossinghoffDepartment of MathematicsDavidson CollegeDavidson, North Carolina 28035-6996mjm "at" cecm.sfu.caLast modified August 16, 2005.