EXHIBIT
    31.1   CERTIFICATIONS  I,
    Marvin S. Hausman, M.D., certify that:  1.
    I have reviewed this quarterly report on Form 10-Q of Axonyx Inc.;  2.
    Based on my knowledge, this quarterly report does not contain any untrue statement
    of a material fact or omit to state a material fact necessary to make the
    statements made, in light of the circumstances under which such statements
    were made, not misleading with respect to the period covered by this quarterly
    report;  3.
    Based on my knowledge, the financial statements, and other financial information
    included in this quarterly report, fairly present in all material respects
    the financial condition, results of operations and cash flows of the registrant
    as of, and for, the periods presented in this quarterly report;  4.
    The registrants other certifying officer and I are responsible for establishing
    and maintaining disclosure controls and procedures (as defined in Exchange
    Act Rules 13a-14 and 15d-14) for the registrant and we have:  a)
    designed such disclosure controls and procedures, or caused such disclosure
    controls and procedures to be designed under our supervision, to ensure that
    material information relating to the registrant, including its consolidated
    subsidiaries, is made known to us by others within those entities, particularly
    during the period in which this quarterly report is being prepared; and  b)
    evaluated the effectiveness of the registrants disclosure controls and
    procedures and presented in this report our conclusions about the effectiveness
    of the disclosure controls and procedures, as of the end of the period covered
    by this quarterly report based on such evaluation (the Evaluation Date);
    and  c)
    disclosed in this report any change in the registrants internal control
    over financial reporting that occurred during the registrants most recent
    fiscal quarter that has materially affected, or is reasonably likely to affect,
    the registrants internal control over financial reporting.  5.
    The registrants other certifying officers and I have disclosed, based
    on our most recent evaluation, to the registrants auditors and the audit
    committee of registrants board of directors (or persons performing the
    equivalent function):  a)
    all significant deficiencies in the design or operation of internal controls
    which are reasonably likely to adversely affect the registrants ability
    to record, process, summarize and report financial data and have identified
    for the registrants auditors any material weaknesses in internal controls;
    and  b)
    any fraud, whether or not material, that involves management or other employees
    who have a significant role in the registrants internal controls; and  6.
    The registrants other certifying officer and I have indicated in this
    quarterly report whether or not there were significant changes in internal
    controls or in other factors that could significantly affect internal controls
    subsequent to the date of our most recent evaluation, including any corrective
    actions with regard to significant deficiencies and material weaknesses.  Date:
    November 10, 2004   /s/
          Marvin S. Hausman, M.D.
Marvin
    S. Hausman, M.D.
  Chairman
    and Chief Executive Officer