<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2016-01-12</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000872589</issuerCik>
        <issuerName>REGENERON PHARMACEUTICALS INC</issuerName>
        <issuerTradingSymbol>REGN</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001121404</rptOwnerCik>
            <rptOwnerName>Sanofi</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>54, RUE LA BOETIE</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>PARIS</rptOwnerCity>
            <rptOwnerState>I0</rptOwnerState>
            <rptOwnerZipCode>75008</rptOwnerZipCode>
            <rptOwnerStateDescription>FRANCE</rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2016-01-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>5007</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>453.8912</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>23113577</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2016-01-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
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                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
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                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
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                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
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                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
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                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
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                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
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                <value>Common Stock</value>
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            <transactionDate>
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            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
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                <transactionCode>P</transactionCode>
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            <transactionAmounts>
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                    <value>2164</value>
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                </sharesOwnedFollowingTransaction>
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                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
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                <value>Common Stock</value>
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                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
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                    <footnoteId id="F2"/>
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            <postTransactionAmounts>
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                    <value>See note</value>
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                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
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                <value>Common Stock</value>
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            </transactionCoding>
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                <transactionPricePerShare>
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                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
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                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
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                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2016-01-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
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                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
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                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
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                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
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                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
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                <value>2016-01-12</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
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                <transactionCode>P</transactionCode>
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            </transactionCoding>
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                <transactionAcquiredDisposedCode>
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                    <value>See note</value>
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                </natureOfOwnership>
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        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
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            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
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                </natureOfOwnership>
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            <securityTitle>
                <value>Common Stock</value>
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                </natureOfOwnership>
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                    <value>See note</value>
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                <transactionShares>
                    <value>3563</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F2"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>472.6410</value>
                    <footnoteId id="F1"/>
                    <footnoteId id="F21"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>23192508</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See note</value>
                    <footnoteId id="F4"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">The number of securities reported represents an aggregate number of shares purchased in multiple market transactions over a range of purchase prices. The price reported represents the weighted average price per share. The Reporting Person undertakes to provide the staff of the SEC, the Issuer, or a stockholder of the Issuer, upon request, the number of shares purchased by the Reporting Person at each separate price within the range.</footnote>
        <footnote id="F2">Represents shares acquired directly by sanofi-aventis Amerique du Nord (&quot;SAAN&quot;).</footnote>
        <footnote id="F3">Purchase prices range from $453.09 to $454.07 per share, inclusive.</footnote>
        <footnote id="F4">Indirectly owned through (a) SAAN, a direct, wholly-owned subsidiary of Sanofi, and (b) Aventis Pharmaceuticals Inc. (&quot;Aventis&quot;), an indirect, wholly-owned subsidiary of SAAN.  After giving effect to all acquisitions reported on this Form 4, the number of shares beneficially owned directly by SAAN and Aventis was 20,392,956 shares and 2,799,552 shares, respectively.  Pursuant to the Amended and Restated Investor Agreement, dated as of January 11, 2014, by and among Sanofi, SAAN, sanofi-aventis US LLC, Aventis (collectively, the &quot;Sanofi Parties&quot;) and the Issuer, the Sanofi Parties have agreed to vote their respective shares of the Issuer, subject to specified exceptions, in accordance with the recommendation of the Issuer's Board of Directors.</footnote>
        <footnote id="F5">Purchase prices range from $454.21 to $455.19 per share, inclusive.</footnote>
        <footnote id="F6">Purchase prices range from $455.24 to $456.19 per share, inclusive.</footnote>
        <footnote id="F7">Purchase prices range from $456.59 to $457.54 per share, inclusive.</footnote>
        <footnote id="F8">Purchase prices range from $457.60 to $458.41 per share, inclusive.</footnote>
        <footnote id="F9">Purchase prices range from $459.64 to $460.52 per share, inclusive.</footnote>
        <footnote id="F10">Purchase prices range from $460.71 to $461.66 per share, inclusive.</footnote>
        <footnote id="F11">Purchase prices range from $461.73 to $462.71 per share, inclusive.</footnote>
        <footnote id="F12">Purchase prices range from $462.83 to $463.46 per share, inclusive.</footnote>
        <footnote id="F13">Purchase prices range from $463.96 to $464.92 per share, inclusive.</footnote>
        <footnote id="F14">Purchase prices range from $464.99 to $465.93 per share, inclusive.</footnote>
        <footnote id="F15">Purchase prices range from $465.99 to $466.97 per share, inclusive.</footnote>
        <footnote id="F16">Purchase prices range from $466.99 to $467.94 per share, inclusive.</footnote>
        <footnote id="F17">Purchase prices range from $468.09 to $468.87 per share, inclusive.</footnote>
        <footnote id="F18">Purchase prices range from $469.11 to $470.10 per share, inclusive.</footnote>
        <footnote id="F19">Purchase prices range from $470.11 to $471.10 per share, inclusive.</footnote>
        <footnote id="F20">Purchase prices range from $471.14 to $471.92 per share, inclusive.</footnote>
        <footnote id="F21">Purchase prices range from $472.31 to $473.23 per share, inclusive.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ John Felitti, Associate Vice President, Corporate Law, Financial &amp; Securities Law</signatureName>
        <signatureDate>2016-01-14</signatureDate>
    </ownerSignature>
</ownershipDocument>
